CBSE Class 10 Answered

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(1)     When considering two consecutive positive numbers, one of them will always be an odd number and the other will be even as odd and even numbers exist in pairs of consecutive numbers. Therefore, by Euler’s division lemma, the product of two consecutive positive numbers will always be divisible by 2 because all even numbers have 2 as their factor.

(2)     If two numbers have 16 as their HCF and 380 as their LCM, then those numbers must be divisible by 16 and must also divide 380. Concluding from the two statements, 16 must divide 380. However, when divided by 16, 380 leave a remainder 12. Hence, no two numbers can have 16 as HCF and 380 as LCM simultaneously.

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