Do you see that, in each case, the number by which the LCM. Class 7

Do you see that, in each case, the number by which the LCM. Class 7

Question 1.

Do you see that, in each case, the number by which the LCM is multiplied to get the product is actually the HCF? Class 7

Solution:

Yes.

Question 2.

Why does this happen? Can you give an explanation or proof? Class 7

[Hint: Consider the prime factorisation of the given numbers. Among their prime factors, some are common to both factorisations, and the rest occur in only one of them. Between the HCF and the LCM, see how the common and non-common prime factors get distributed. In the product, observe how these two kinds of prime factors occur. Compare them.]

Solution:

Same common factors of the two numbers are included in both HCF and LCM and hence we get the property.

Question 3.

Explore whether this property holds when 3 numbers are considered. Class 7

Solution:

No. HCF of 2, 3 and 4 is 1, while LCM = 12. HCF × LCM = 1 × 12 = 12 ≠ Product 2 × 3 × 4 = 24.