Quantitative Aptitude PYQs

Shortcut Trick

Use the multiplying factor method: Final Price = Initial Price × (1 − r₁) × (1 − r₂) × (1 − r₃) × (1 + r₄).

Net multiplier = 0.60 × 0.85 × 0.55 × 1.70 = 0.47685.

Net change = (1 − 0.47685) × 100 = 52.315% decrease.

∴ The correct answer is Decreases by 52.32%.

Alternate Method

Given: Initial Price = 100, Successive Decreases = 40%, 15%, 45%, Increase = 70%.

⇒ Price after 1^st month = 100 × (60/100) = 60

⇒ Price after 2^nd month = 60 × (85/100) = 51

⇒ Price after 3^rd month = 51 × (55/100) = 28.05

⇒ Price after 4^th month = 28.05 × (170/100) = 47.685

⇒ Total decrease = 100 − 47.685 = 52.315%

⇒ Rounding off to two decimal places = 52.32%

∴ The correct answer is Decreases by 52.32%.

Additional Information

Successive Percentage Change

For two changes a% and b%, the net change is [a + b + (a × b)/100]%. Use negative for decrease.

Multiplying Factor

A decrease of x% is equivalent to multiplying the value by (1 − x/100), while an increase is (1 + x/100).

Base Value Selection

Always assume 100 as the initial value in percentage problems to simplify calculations and direct conversion to %.

Shortcut Trick

Time taken for 1 page = 40 ÷ 4 = 10 minutes.

Pages typed in 90 minutes = 90 ÷ 10 = 9 pages.

∴ The correct answer is 9 pages.

Alternate Method

Given:

Time₁ = 40 minutes, Work₁ = 4 pages

Time₂ = 90 minutes, Work₂ = x pages

Formula Used:

Using the Work and Time relationship: (W₁ / T₁) = (W₂ / T₂)

Calculations:

⇒ 4 / 40 = x / 90

⇒ 1 / 10 = x / 90

⇒ x = 90 ÷ 10

⇒ x = 9

∴ The correct answer is 9 pages.

Additional Information

Direct Proportion

When one quantity increases, the other also increases. Formula: x₁/y₁ = x₂/y₂.

Unitary Method

Find the value of a single unit first, then multiply by the required number of units.

Efficiency and Time

Work = Efficiency × Time. Here, typing speed is the constant efficiency.

Shortcut Trick

Age of new member = New Average + (Initial Count × Increase in Average)

Increase in average = 26 − 25 = 1 year

Age of teacher = 26 + (23 × 1)

Age of teacher = 26 + 23 = 49 years

∴ The correct answer is 49 years.

Alternate Method

Given:

Initial number of students = 23; Initial average age = 25 years

New number of people (students + teacher) = 24; New average age = 26 years

Formula: Total Age = Number of People × Average Age

⇒ Total age of 23 students = 23 × 25 = 575 years

⇒ Total age of 24 people (including teacher) = 24 × 26 = 624 years

⇒ Age of the teacher = Total age of 24 people − Total age of 23 students

⇒ Age of the teacher = 624 − 575 = 49 years

∴ The correct answer is 49 years.

Additional Information

Inclusion in Average

When a new member joins, their value = New Average + Old Count × (New Average − Old Average).

Exclusion from Average

When a member leaves, their value = Old Average + New Count × (Old Average − New Average).

Weighted Average Formula

The combined average of two groups is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average.

Shortcut Trick

The empirical relationship between central tendencies is: Mode − Median = 2/3 (Mode − Mean).

Given that the Mode exceeds the Mean by 19.8, we have: (Mode − Mean) = 19.8.

Substituting the value: Required difference = 2/3 × 19.8.

Required difference = 2 × 6.6 = 13.2.

∴ The correct answer is 13.2.

Alternate Method

Given:

Mode − Mean = 19.8

Formula Used:

The Empirical Formula: Mode = 3 Median − 2 Mean

Calculations:

⇒ From the formula: Mode = 3 Median − 2 Mean

⇒ Mode − Mean = (3 Median − 2 Mean) − Mean

⇒ Mode − Mean = 3 Median − 3 Mean = 3 (Median − Mean)

⇒ 19.8 = 3 (Median − Mean)

⇒ Median − Mean = 19.8 ÷ 3 = 6.6

⇒ Now, Mode − Median = (Mode − Mean) − (Median − Mean)

⇒ Mode − Median = 19.8 − 6.6

⇒ Mode − Median = 13.2

∴ The correct answer is 13.2.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is Mode = 3 Median − 2 Mean.

Symmetric Distribution

In a perfectly symmetrical distribution (like the Normal Distribution), the values of Mean, Median, and Mode are equal: Mean = Median = Mode.

Positively Skewed Distribution

When the tail is on the right side, the relationship is typically Mean > Median > Mode.

Negatively Skewed Distribution

When the tail is on the left side, the relationship is typically Mode > Median > Mean.

Shortcut Trick

Speed Ratio = 5 : 3 (Normal : New)

Time Ratio = 3 : 5 (Inverse of speed as distance is constant)

Difference in time units = 5 − 3 = 2 units

2 units = 24 minutes ⇒ 1 unit = 12 minutes

Usual Time = 3 units = 3 × 12 minutes = 36 minutes

∴ The correct answer is 36 min.

Alternate Method

Given: New speed = 3/5 of Normal speed, Late time = 24 minutes.

Formula Used: Usual Time = [Late Time × Numerator] ÷ [Denominator − Numerator]

⇒ Let the usual time be T.

⇒ New time = T + 24

⇒ Since Distance = Speed × Time, S × T = (3/5)S × (T + 24)

⇒ T = 3/5 (T + 24)

⇒ 5T = 3T + 72

⇒ 2T = 72

⇒ T = 36 minutes

∴ The correct answer is 36 min.

Additional Information

Speed-Time Inverse Proportionality

When distance is constant, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Late/Early Logic

If a person is late by 'L' minutes at a/b of usual speed, Usual Time = [a × L] / |b − a|.

Relative Speed

Used for two moving bodies; add speeds if moving in opposite directions, subtract if in the same direction.

Shortcut Trick

Effective rate for 10% compound interest for 2 years = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = Principal × (100% + Effective Rate%)

⇒ Amount = 1,500 × 121%

⇒ Amount = 15 × 121 = 1,815.

∴ The correct answer is ₹1,815.

Alternate Method

Given:

Principal (P) = ₹1,500

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 1,500 × (1 + 10/100)²

⇒ A = 1,500 × (110/100) × (110/100)

⇒ A = 1,500 × (11/10) × (11/10)

⇒ A = 1,500 × 121/100

⇒ A = 15 × 121

⇒ A = 1,815

∴ The correct answer is ₹1,815.

Additional Information

Compound Interest Formula

Compound Interest (CI) is calculated as CI = A − P, where A = P(1 + r/n)^nt.

Difference between CI and SI for 2 Years

The difference (D) for two years is given by D = P(R/100)².

Successive Percentage for 2 Years

The net effective rate for two successive increases of a% and b% is (a + b + ab/100)%.

Cross multiply ⇒ X × X = 125 × 5

⇒ X² = 625

⇒ X = √625 = 25 (since X > 0)

∴ Answer = 25

Given:

X ÷ 125 = 5 ÷ X

Formula used:

If a/b = c/d ⇒ ad = bc

Calculation:

⇒ X/125 = 5/X

⇒ X × X = 125 × 5

⇒ X² = 625

⇒ X = ±25

⇒ Since X > 0 ⇒ X = 25

∴ The correct answer is 25.

Cross Multiplication

If a/b = c/d then a × d = b × c

Quadratic Equation

x² = k ⇒ x = ±√k

Condition Check

Given X > 0, so only positive root is valid

Perfect Square

625 = 25²

Shortcut Trick

Product of fractions: (40/100) × (10/100) × (1/3) × Number = 477

(2/5) × (1/10) × (1/3) × Number = 477

(1/75) × Number = 477

Number = 477 × 75 = 35775

∴ The correct answer is 35775.

Alternate Method

Given: 40% of 10% of 1/3 of x = 477

⇒ Let the number be x.

⇒ (40/100) × (10/100) × (1/3) × x = 477

⇒ (0.4) × (0.1) × (1/3) × x = 477

⇒ (0.04 / 3) × x = 477

⇒ x = (477 × 3) / 0.04

⇒ x = 1431 / 0.04

⇒ x = 35775

∴ The correct answer is 35775.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 40% = 2/5, 10% = 1/10, 25% = 1/4.

Successive Percentage Rule

A% of B% of C is equivalent to (A × B / 100)% of C. Use this for multi-step percentage problems.

Reverse Calculation

When the final value is given, the original number = Result ÷ (Product of all percentage/fraction multipliers).

Shortcut Trick

The best scheme for the shopkeeper is the one that offers the minimum discount to the customer.

Scheme A: 32 + 40 − (32 × 40) ÷ 100 = 72 − 12.8 = 59.2%

Scheme B: 5 ÷ (1 + 5) × 100 = 5 ÷ 6 × 100 = 83.33%

Scheme C: Single discount = 48%

Scheme D: 8 + 33 − (8 × 33) ÷ 100 = 41 − 2.64 = 38.36%

Comparing all, Scheme D provides the lowest discount (38.36%), making it the most profitable for the seller.

∴ The correct answer is D.

Alternate Method

Given → Successive discounts and Buy-Get schemes.
Formulas: Successive Discount = a + b − (ab ÷ 100) | Buy X Get Y free Discount% = [Y ÷ (X + Y)] × 100.

⇒ Calculate effective discount for Scheme A: 32 + 40 − 12.8 = 59.2%.
⇒ Calculate effective discount for Scheme B: (5 ÷ 6) × 100 = 83.33%.
⇒ Calculate effective discount for Scheme C: 48%.
⇒ Calculate effective discount for Scheme D: 8 + 33 − (264 ÷ 100) = 41 − 2.64 = 38.36%.
⇒ A shopkeeper always prefers the lowest discount rate to maximize the Selling Price.
⇒ Since 38.36% < 48% < 59.2% < 83.33%, Scheme D is best.

∴ The correct answer is D.

Additional Information

Best for Customer vs Shopkeeper

The Best Scheme for Customer is the one with the Maximum Discount. Conversely, the Best Scheme for Shopkeeper is the one with the Minimum Discount.

Buy X Get Y Free

In a 'Buy X, Get Y free' offer, the total number of articles given is (X + Y). The discount percentage is calculated on the total articles: [Free Articles ÷ Total Articles] × 100.

Shortcut Trick

For any quadrilateral ABCD, the angle formed by the bisectors of two adjacent angles (say ∠C and ∠D) is always equal to half the sum of the other two angles (∠A and ∠B):

∠DOC = (∠A + ∠B) ÷ 2

Substituting the given values:

∠DOC = (32° + 58°) ÷ 2 = 90° ÷ 2 = 45°

∴ The correct answer is 45°.

Alternate Method

Given:

In quadrilateral ABCD:

∠A = 32°

∠B = 58°

DO and CO are the angle bisectors of ∠D and ∠C respectively, meeting at point O.

Formula Used:

The sum of all interior angles of a quadrilateral is 360°.

The sum of all interior angles of a triangle is 180°.

Calculations:

In quadrilateral ABCD:

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ 32° + 58° + ∠C + ∠D = 360°

⇒ 90° + ∠C + ∠D = 360°

⇒ ∠C + ∠D = 360° − 90°

⇒ ∠C + ∠D = 270°

Dividing the entire equation by 2:

⇒ (∠C + ∠D) ÷ 2 = 270° ÷ 2

⇒ ∠C/2 + ∠D/2 = 135°

Since CO and DO are the angle bisectors of ∠C and ∠D:

⇒ ∠OCD = ∠C/2 and ∠ODC = ∠D/2

⇒ ∠OCD + ∠ODC = 135°

Now, in triangle DOC, the sum of all interior angles is 180°:

⇒ ∠DOC + ∠OCD + ∠ODC = 180°

⇒ ∠DOC + 135° = 180°

⇒ ∠DOC = 180° − 135°

⇒ ∠DOC = 45°

∴ The correct answer is 45°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles: ∠DOC = (∠A + ∠B) ÷ 2.

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral (say ∠A and ∠C) intersect, the angle between them is equal to half the absolute difference of the other two angles: |∠B − ∠D| ÷ 2.

Sum of Interior Angles of a Polygon

The sum of the interior angles of any convex polygon with n sides is (n − 2) × 180°. For a quadrilateral, n = 4, so the sum is 360°.

Shortcut Trick

Total population = 328000, and Male population = 164000.

Female population = 328000 − 164000 = 164000.

Since the number of males equals females, the ratio of their population is 1 :

1. When quantities are equal, the overall literacy rate is the simple average of male and female literacy rates.

(34% + Female Literacy%) ÷ 2 = 24% ⇒ Female Literacy% = 48% − 34% = 14%.

∴ The correct answer is 14%.

Alternate Method

Given:

Total population = 328000

Number of males = 164000

Overall literacy rate = 24%

Male literacy rate = 34%

34%
Male

x%
Female

24%
Overall

24−x

10

Ratio = 1 : 1
24 − x = 10 ⇒ x = 14%

Calculations:

⇒ Number of females = Total population − Number of males

⇒ Number of females = 328000 − 164000 = 164000

⇒ Total literate population = 24% of 328000

⇒ Total literates = (24 ÷ 100) × 328000 = 78720

⇒ Number of literate males = 34% of 164000

⇒ Literate males = (34 ÷ 100) × 164000 = 55760

⇒ Number of literate females = Total literates − Literate males

⇒ Literate females = 78720 − 55760 = 22960

⇒ Percentage of literate females = (Literate females ÷ Total females) × 100

⇒ Percentage = (22960 ÷ 164000) × 100 = 14%

∴ The correct answer is 14%.

Additional Information

Alligation Method

When two groups are mixed to form a whole, the ratio of their quantities is inversely proportional to the differences between their averages and the overall average. Q₁ ÷ Q₂ = (A₂ − A_overall) ÷ (A_overall − A₁).

Weighted Average

The combined average of multiple sub-groups is calculated using the formula: A = (N₁A₁ + N₂A₂) ÷ (N₁ + N₂), where N denotes the respective count and A denotes the specific average.

Percentages Concept

A percentage is a fraction out of 100. When calculating the percentage value of a sub-group, the denominator must strictly be the size of that specific sub-group, not the overall size.

Shortcut Trick

Divisibility by 11 requires the difference between the sum of alternate digits to be 0 or a multiple of 11.

Sum of odd-placed digits = N + 5 + 5 = N + 10.

Sum of even-placed digits = 7 + M + 2 = M + 9.

Difference = (N + 10) − (M + 9) = N − M + 1.

Equating the difference to 0 gives: N − M + 1 = 0 ⇒ M − N = 1.

∴ The correct answer is M − N = 1.

Alternate Method

Given:

The 6-digit number N75M52 is divisible by 11.

Formula Used:

A number is divisible by 11 if the difference of the sums of its alternate digits is 0 or a multiple of 11.

Calculations:

⇒ Sum of digits at odd positions (1^st, 3^rd, 5^th from left) = N + 5 + 5 = N + 10

⇒ Sum of digits at even positions (2^nd, 4^th, 6^th from left) = 7 + M + 2 = M + 9

⇒ Difference = (Sum of odd positions) − (Sum of even positions)

⇒ Difference = (N + 10) − (M + 9)

⇒ Difference = N − M + 1

⇒ For divisibility by 11, the difference must be 0, 11, −11, etc.

⇒ Case 1: Equating the difference to 0

⇒ N − M + 1 = 0 ⇒ M − N = 1

⇒ Case 2: Equating the difference to 11

⇒ N − M + 1 = 11 ⇒ N − M = 10 (Not possible as N and M must be single digits between 0 and 9)

⇒ Therefore, the only valid relation is M − N = 1.

∴ The correct answer is M − N = 1.

Additional Information

Divisibility Rule of 3 & 9

A number is divisible by 3 or 9 if the sum of its digits is exactly divisible by 3 or 9 respectively.

Divisibility Rule of 4 & 8

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if the last three digits form a number divisible by 8.

Divisibility Rule of 7

Double the last digit and subtract it from the remaining truncated number. If the result is 0 or divisible by 7, the original number is divisible by 7.

Shortcut Trick

When a number is decreased by 6%, it becomes 94% of its original value.

⇒ 94% of the number = 3760

⇒ 1% of the number = 3760 ÷ 94 = 40

⇒ 100% of the number = 40 × 100 = 4000

∴ The correct answer is 4000.

Alternate Method

Original Number (100%)

Remaining: 94% = 3760
6%

Let the number be x

x − 6% of x = 3760

Given:

A number decreased by 6% is equal to 3760.

Calculations:

Let the original number be x.

⇒ x − (6% of x) = 3760

⇒ x − (6x / 100) = 3760

⇒ (100x − 6x) / 100 = 3760

⇒ 94x / 100 = 3760

⇒ x = (3760 × 100) / 94

⇒ x = 40 × 100

⇒ x = 4000

∴ The correct answer is 4000.

Additional Information

Percentage Increase / Decrease

When a value is increased by R%, the new value becomes (100 + R)% of the original. When it is decreased by R%, it becomes (100 − R)% of the original value.

Reverse Percentage Calculation

To find the original value before a percentage change, divide the final value by the multiplying factor. Original = Final Value ÷ (1 ± R/100).

Successive Percentage Change

If a number is successively changed by A% and B%, the net overall percentage change is calculated as: A + B + (A × B)/100.

Shortcut Trick

The third proportional of two numbers a and b is calculated directly using the formula: x = b² ÷ a

Given the numbers: a = 9, b = 60

x = (60 × 60) ÷ 9

x = 3600 ÷ 9 = 400

∴ The correct answer is 400.

Alternate Method

Given:

First number = 9

Second number = 60

Third proportional = x

Concept Used:

If three numbers are in continued proportion, then the ratio of the first to the second is equal to the ratio of the second to the third (a : b :: b : x).

Product of Extremes (9 × x)
Product of Means (60 × 60)

9
60
60
x

:
::
:

9 × x = 60 × 60 ⇒ x = 400

Calculations:

⇒ 9 : 60 :: 60 : x

⇒ 9 ÷ 60 = 60 ÷ x

⇒ 9 × x = 60 × 60

⇒ 9x = 3600

⇒ x = 3600 ÷ 9

⇒ x = 400

∴ The correct answer is 400.

Additional Information

Mean Proportional

The mean proportional between two quantities a and b is geometrically represented as √(a × b).

Fourth Proportional

If a : b :: c : d, then d is termed as the fourth proportional to a, b, and c. It is calculated using d = (b × c) ÷ a.

Continued Proportion

Three numbers a, b, and c are said to be in continued proportion if a : b = b : c. Here, b forms the mean proportional while c remains the third proportional.

Shortcut Trick

The product of two consecutive numbers is slightly greater than the smaller number squared.

Find the closest integer to √552, which is approximately 23.49.

The consecutive natural numbers must be 23 and 24.

Checking the product: 23 × 24 = 552.

The greater of the two numbers is 24.

∴ The correct answer is 24.

Alternate Method

Given:

The product of two consecutive natural numbers = 552

Concept Used:

Let the two consecutive natural numbers be x and (x + 1).

Calculations:

⇒ x × (x + 1) = 552

⇒ x² + x = 552

⇒ x² + x − 552 = 0

⇒ x² + 24x − 23x − 552 = 0

⇒ x(x + 24) − 23(x + 24) = 0

⇒ (x − 23)(x + 24) = 0

⇒ x = 23 or x = −24

⇒ Since x must be a natural number, x = 23.

⇒ The consecutive numbers are 23 and 24.

⇒ The greater number = x + 1 = 23 + 1 = 24.

∴ The correct answer is 24.

Additional Information

Quadratic Equation Roots

The roots of any standard quadratic equation ax² + bx + c = 0 can be found by factorisation (splitting the middle term) or by using the formula x = (−b ± √(b² − 4ac)) ÷ 2a.

Properties of Consecutive Numbers

The product of two consecutive integers n and (n + 1) is always an even number and perfectly matches the mathematical identity n² + n.

Natural Numbers

Natural numbers represent the set of positive counting integers starting from 1 (1, 2, 3, ...), thereby excluding zero, fractions, and negative integers.

Shortcut Trick

Observe the unit digit of the given number 630605.

The unit digit is 5.

According to the divisibility rules, any number with a unit digit of 0 or 5 is completely divisible by 5.

∴ The correct answer is 5.

Alternate Method

Given:

The number to be divided is 630605.

Calculations:

Let us test the divisibility of 630605 by all the given options to confirm.

⇒ Divisibility by 3: Sum of the digits = 6 + 3 + 0 + 6 + 0 + 5 = 20.

⇒ Since 20 is not divisible by 3, the number is not divisible by 3.

⇒ Divisibility by 5: The unit digit of 630605 is 5. Any number ending with 5 is exactly divisible by 5.

⇒ Divisibility by 7: 630605 ÷ 7 = 90086 with a remainder of 3. Hence, it is not divisible by 7.

⇒ Divisibility by 14: An odd number (ending in 5) can never be divisible by an even number like 14.

⇒ Therefore, 630605 is only divisible by 5 among the given choices.

∴ The correct answer is 5.

Additional Information

Divisibility by 3 & 9

A number is divisible by 3 or 9 if the sum of all its digits is exactly divisible by 3 or 9, respectively.

Divisibility by 4 & 8

A number is divisible by 4 if its last two digits are divisible by 4. It is divisible by 8 if its last three digits are divisible by 8.

Divisibility by 7

Double the last digit and subtract it from the remaining truncated number. If the result is exactly divisible by 7, the original number is also divisible by 7.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

Shortcut Trick

An equilateral triangle has 3 equal sides.

Perimeter = 3 × Side

Perimeter = 3 × 13 = 39 cm

∴ The correct answer is 39 cm.

Alternate Method

A
B
C
13 cm
13 cm
13 cm

Given:

Side of the equilateral triangle (a) = 13 cm

Formula Used:

Perimeter of an equilateral triangle = a + a + a = 3a

Calculation:

⇒ Perimeter = 3 × 13

⇒ Perimeter = 39 cm

∴ The correct answer is 39 cm.

Additional Information

Area of Equilateral Triangle

The area is calculated using the formula: Area = (√3 ÷ 4) × a², where a is the side length.

Height of Equilateral Triangle

The altitude or height is given by: h = (√3 ÷ 2) × a.

Key Properties

All three sides are equal in length, and all three internal angles measure exactly 60°. The centroid, circumcenter, incenter, and orthocenter all coincide at the same central point.

Shortcut Trick

Convert all given terms to prime bases to find the accurate LCM.

T1 = 3³ × 2⁶ × 13, T2 = 3² × 13² × 2⁴, T3 = 2⁹ × 13² × 2⁸ = 2¹⁷ × 13².

LCM = Product of highest powers of all prime factors = 2¹⁷ × 3³ × 13².

Checking Option 1: 3³ × 8³ × 13² × 16² = 3³ × 2⁹ × 13² × 2⁸ = 2¹⁷ × 3³ × 13².

∴ The correct answer is 3³ × 8³ × 13² × 16².

Alternate Method

Given:

First Expression = 3³ × 8² × 13

Second Expression = 3² × 13² × 16

Third Expression = 8³ × 13² × 16²

Formula Used:

The LCM of numbers in exponential form is the product of the highest powers of all prime factors present in the numbers.

Calculations:

⇒ Prime factorizing bases 8 and 16: 8 = 2³ and 16 = 2⁴

⇒ Rewrite First Expression: 3³ × (2³)² × 13 = 3³ × 2⁶ × 13¹

⇒ Rewrite Second Expression: 3² × 13² × 2⁴ = 3² × 2⁴ × 13²

⇒ Rewrite Third Expression: (2³)³ × 13² × (2⁴)² = 2⁹ × 13² × 2⁸ = 2¹⁷ × 13²

⇒ The unique prime factors are 2, 3, and 13.

⇒ Highest power of 2 = 2¹⁷

⇒ Highest power of 3 = 3³

⇒ Highest power of 13 = 13²

⇒ LCM = 2¹⁷ × 3³ × 13²

⇒ Now, check the given options to find the matching value.

⇒ Option 1: 3³ × 8³ × 13² × 16²

⇒ 3³ × (2³)³ × 13² × (2⁴)² = 3³ × 2⁹ × 13² × 2⁸ = 2¹⁷ × 3³ × 13²

⇒ This matches our calculated LCM exactly.

∴ The correct answer is 3³ × 8³ × 13² × 16².

Additional Information

HCF of Exponential Numbers

The HCF (Highest Common Factor) of numbers in exponential form is the product of the lowest powers of all common prime factors.

LCM of Fractions

The LCM of given fractions is calculated using the relation: (LCM of Numerators) ÷ (HCF of Denominators).

Product Rule of Indices

When multiplying expressions with the same base, their exponents are added together according to the formula a^m × aⁿ = a^m+n.

Shortcut Trick

Sum of digits of 253118139 = 2 + 5 + 3 + 1 + 1 + 8 + 1 + 3 + 9 = 33.

Since 33 is a multiple of 3, the number is divisible by 3.

The number ends in an odd digit (9), so it cannot be divisible by even numbers like 8 or 12.

It does not end in 0 or 5, so it is not divisible by 5.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 253118139

Method Used:

Check the divisibility rules for each given option.

Calculations:

⇒ Divisibility by 3: Sum of all digits = 2 + 5 + 3 + 1 + 1 + 8 + 1 + 3 + 9 = 33. Since 33 ÷ 3 = 11, the number is exactly divisible by 3.

⇒ Divisibility by 12: A number must be divisible by both 3 and 4 to be divisible by 12. The last two digits are 39, which is not divisible by 4. So, it is not divisible by 12.

⇒ Divisibility by 5: The last digit must be 0 or 5. Here, the last digit is 9, so it is not divisible by 5.

⇒ Divisibility by 8: The last three digits must be divisible by 8. 139 is an odd number and therefore not divisible by 8.

∴ The correct answer is 3.

Additional Information

Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

Divisibility by 4 and 8

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. It is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility by 6

A number is divisible by 6 if it satisfies the divisibility rules of both 2 (even last digit) and 3 (sum of digits divisible by 3).

Shortcut Trick

The direct formula for the average of the first n even numbers is (n + 1).

Here, the total number of terms, n = 169.

⇒ Average = 169 + 1

⇒ Average = 170

∴ The correct answer is 170.

Alternate Method

Given:

Number of terms (n) = 169

The sequence consists of the first 169 positive even numbers.

Average of an Arithmetic Progression

2
1^st Term

170
Average

338
169^th Term
Average = (First Term + Last Term) ÷ 2

Formula Used:

⇒ Last term of the even number series = 2 × n

⇒ Average of an Arithmetic Progression = (First Term + Last Term) ÷ 2

Calculations:

⇒ First even number = 2

⇒ 169^th even number = 169 × 2 = 338

⇒ Average = (2 + 338) ÷ 2

⇒ Average = 340 ÷ 2

⇒ Average = 170

∴ The correct answer is 170.

Additional Information

Average of First n Natural Numbers

The average of the first n natural numbers is given by the formula: (n + 1) ÷ 2.

Average of First n Odd Numbers

The average of the first n consecutive odd numbers is exactly equal to n.

Average of an Arithmetic Progression

For any uniform sequence that forms an arithmetic progression, the average is always the mean of its first and last terms: (First Term + Last Term) ÷ 2.

Average of Squares of First n Natural Numbers

The average of the squares of the first n natural numbers is: (n + 1)(2n + 1) ÷ 6.