Shortcut Trick
The ratio of HCF to LCM is 1 : 6, so total parts = 1 + 6 = 7.
7 parts = 189 ⇒ 1 part (HCF) = 27 and 6 parts (LCM) = 162.
Using the rule: Number1 × Number2 = LCM × HCF
54 × Number2 = 162 × 27
Number2 = (162 × 27) ÷ 54 = 81
∴ The correct answer is 81.
Alternate Method
Given:
LCM = 6 × HCF
LCM + HCF = 189
First Number = 54
Formula Used:
LCM × HCF = Product of the two numbers
Calculations:
⇒ Let the HCF be x.
⇒ LCM = 6x
⇒ LCM + HCF = 189
⇒ 6x + x = 189
⇒ 7x = 189
⇒ x = 189 ÷ 7 = 27
⇒ So, HCF = 27 and LCM = 6 × 27 = 162
⇒ Now, First Number × Second Number = LCM × HCF
⇒ 54 × Second Number = 162 × 27
⇒ Second Number = (162 × 27) ÷ 54
⇒ Second Number = 162 × (27 ÷ 54)
⇒ Second Number = 162 ÷ 2 = 81
∴ The correct answer is 81.
Additional Information
Product Rule of HCF and LCM
For any two positive integers a and b, the product of the numbers is equal to the product of their HCF and LCM. Formula: a × b = HCF(a, b) × LCM(a, b).
Co-prime Property
If two numbers are expressed as H × x and H × y (where H is their HCF), then x and y must be co-prime to each other. Their LCM is mathematically given by H × x × y.
Fractions HCF and LCM
HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators). LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators).