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Given:

A number, when divided by 987, gives a remainder 59.

Concept used:

Dividend = (Divisor × Quotient) + Remainder

Calculation:

Let the quotient be k.

According to the above concept,

⇒ Dividend = (987 × k) + 59

⇒ 987k + 59

Now, when the number is divided by 21

⇒ \(\frac{987k + 59}{{21}}\) = 47k + \(\frac{59}{{21}}\)

While solving we get quotient as 2 and remainder as 17.

∴ The required remainder is 17.

Concept used:

Positive integer: Positive integers are the numbers we use to count: 1, 2, 3, 4 and so on.

Euclid division lemma: If we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.

Calculation:

According to the above concept,

Positive integer = 143 × k + 28  

⇒ 13 × 11 × k + 13 × 2 + 2

⇒ 13(11k + 2) + 2

So, required remainder = 2

∴ The remainder obtained on dividing the same number by 13 is 2.

Given:

(Np-1 - 1) is a multiple of p

Concept used:

Prime number: A number which is divisible by one and itself.

Co-prime number: Those number that have only one common factor, namely 1.

Integers: A whole number (not a fractional number) that can be positive, negative, or zero

Calculation:

N is prime to p, that means N and p are two co-prime numbers.

Since, (Np-1 - 1) is a multiple of p.

So, (Np-1 - 1) = (k × p)

⇒ k = (Np-1 - 1)/p

Let p = 5 and N = 2  [co-prime numbers with p being a prime number]

So, k = (2^5-1 - 1)/5

⇒ (15/5) = 3  [integers]

That means the given condition satisfies.

Again take p = 4 and N = 3  [co-prime numbers with p being a non prime number]

So, k = (3⁴-1 - 1)/4

⇒ (26/4) = 6.5  [Not a integer]

That means the given condition does not satisfies.

∴ P must be a prime number only.

Given:

LCM of two numbers is 16 times their HCF. 

The sum of LCM and HCF is 850.

If one number is 50

Concept used:

LCM = Product of the numbers/HCF

Calculation:

Let the HCF be x.

Then, LCM = 16x and another number is n.

Now, the sum of the LCM and HCF is

⇒ x + 16x = 850

⇒ 17x = 850

⇒ x = 50

Now, according to the above concept

⇒ 16x = \(\frac{50 \times n}{{x}}\)

⇒ 16 × 50 = \(\frac{50n}{{50}}\)

⇒ n = 800

∴ The other number is 800.

Given:

The HCF of two numbers = 98

LCM = 2352

Concept used:

Co-prime number: 

Coprime numbers are those numbers that do not have any common

factor other than 1.

LCM × HCF = Product of the two numbers

Calculation:

Let the two numbers be 98x and 98y respectively.

Now, according to the above concept,

⇒ (98x) × (98y) = 98 × 2352

⇒ xy = 24

Since co-prime factors of 24 are 3 and 8

Let x = 3 and y = 8

Now, the sum of the number = (98 × 3 + 98 × 8)

⇒ 98(3 + 8) = (98 × 11) = 1078

∴ The required sum of the numbers is 1078.

Given:

The numbers between 500 and 1000 are divisible by 13

Formula used:

_Tn = a + (n - 1)d

Where,

T_n = last number, a = first number,

d = common difference

Calculation:

The first number after 500 and divisible by 13 = 507

And, the number just less than 1000 and divisible by 13 = 988

Hence sequence is

⇒ 507, 520, 533, ......988

Now, according to the above concept,

Tn = a + (n - 1)d

⇒ 988 = 507 + (n - 1)13

⇒ (n - 1) = 481/13

⇒ (n - 1) = 37

⇒ n = 38

∴ Number between 500 and 1000 which is divisible by 13 is 38.

Concept used:

Non-terminating: A decimal number that continues endlessly, with no group of digits repeating endlessly.

Repeating decimal:  A number whose decimal representation eventually becomes periodic (i.e. the same sequence of digits repeats indefinitely).

Calculation:

According to the above options,

⇒ \(\frac{13}{8}\) = 1.625

⇒ \(\frac{3}{16}\) = 0.1875

⇒ \(\frac{3}{11}\) = 0.272727

⇒ \(\frac{137}{25}\) = 5.48

∴ The required non-terminating and repeating decimal is \(\frac{3}{11}\).

Calculation:

According to the given options

⇒ \(\frac{7}{9}\) = 0.77

⇒ \(\frac{11}{14}\) = 0.78

⇒ \(\frac{3}{4}\) = 0.75

⇒ \(\frac{10}{13}\) = 0.76

∴ The largest of the following is \(\frac{11}{14}\).

Shortcut Trick

To find the square root of a surd in the form A + 2√B, find two numbers whose sum is A and product is B.

Here, sum = 9 and product = 14.

The two numbers are 7 and 2 (since 7 + 2 = 9 and 7 × 2 = 14).

Therefore, the square root is simply √7 + √2.

∴ The correct answer is √2 + √7.

Alternate Method

Given: Expression = 9 + 2√14

Formula Used: (a + b)² = a² + b² + 2ab

⇒ Let the square root be (√x + √y).

⇒ (√x + √y)² = x + y + 2√xy

⇒ Comparing x + y + 2√xy with 9 + 2√14:

⇒ x + y = 9 and xy = 14

⇒ 9 + 2√14 = 7 + 2 + 2√(7 × 2)

⇒ 9 + 2√14 = (√7)² + (√2)² + 2(√7)(√2)

⇒ 9 + 2√14 = (√7 + √2)²

⇒ Square root = √(√7 + √2)² = √7 + √2

∴ The correct answer is √2 + √7.

Additional Information

Square Root of a Quadratic Surd

For an expression of the form a ± 2√b, the square root is √m ± √n, where m + n = a and mn = b.

Algebraic Identity for Surds

The identity (a + b)² = a² + b² + 2ab allows us to transform a binomial surd into a perfect square, making it easy to find the square root.

Rationalization

Multiplying a surd like (a + √b) by its conjugate (a − √b) results in a rational number (a² − b).

Concept used:

Perfect square: A number that can be expressed as the product of an integer by itself or as the second exponent of an integer.

Calculation:

∴ The required smallest number is 69, which is added to 1780 to make it perfect square.

Alternate MethodGiven:

The given number = 1780

Calculation:

⇒ **√**1780 = 42.190

Now, finding the square of the next number

⇒ 43² = 1849

Required number = (1849 - 1780) = 69

∴ The required smallest number is 69, which is added to 1780 to make it perfect square.