The sum of two numbers is 1500 and their LCM is 16379

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Answer: Option A : 1489, 11Solution:

Concept:

HCF : The sum of any two numbers is divisible by their HCF. Product of two numbers = HCF × LCM

Given:

Sum of two numbers = 1500 LCM = 16379

Calculation:

Let the two numbers be = a & b ⇒ a + b = 1500 ⇒ b = 1500 - a Let by dividing 1500 by the HCF say h, we get any number, say p. ⇒ 1500 ÷ HCF = p Let by dividing 16379 by HCF say h, we get any number, say q ⇒ 16379 ÷ HCF = q [ p & q are integers] ⇒ h is the HCF of 1500 and 16379 Now, 16379 ÷ 1500 × p = q

1500 = 22 × 3 × 53

16379 = 11 × 1489 Clearly, The HCF (h) of 1500 and 16379 is 1. As we know, a × (1500 - a) = HCF × LCM = 1 × 16379

⇒ 1500a - a2 = 16379

⇒ a2 - 1500a - 16379 = 0
⇒ a2 - 1489a - 11a - 16379 = 0 ⇒ a (a - 1489) - 11(a - 1489) = 0 ⇒ (a - 11) ( a - 1489) = 0 ⇒ a = 11, 1489 For a = 11, b = 1500 - 11 = 1489 For a = 1489, b = 1500 - 1489 = 11

∴ In both the cases, two numbers are 11 and 1489.