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Show that the product of the HCF and LCM of 120 and 126 is equal to the product of 120 and 126.

By looking at their prime factorization, explain why this is so.

Answer :

Final answer:

The product of HCF and LCM of two numbers is equal to the product of those numbers. This is illustrated with the example of 120 and 126, by finding their HCF and LCM through prime factorisation, and showing that these two products are indeed the same.

Explanation:

The subject of this question is the HCF and LCM of two given numbers and analyzing them through their prime factorisation. The HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two numbers have a specific property: the product of the HCF and LCM of two numbers is equal to the product of those two numbers. In the case of 120 and 126:

From these, we can find that the HCF (the product of lowest powers of common primes) is 2 * 3 = 6, and the LCM (the product of highest powers of all primes) is 2³ * 3² * 5 * 7 = 2520.

Now, if we multiply the HCF and LCM we get: 6 * 2520 = 15120, and if we multiply the original numbers, 120 * 126 = 15120. Therefore, the product of HCF and LCM of 120 and 126 is indeed equal to the product of 120 and 126, which signifies the property stated above and explains the inherent connection between the concepts of HCF, LCM and prime factorisation.

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