Multiple choice

The product of two numbers is 1,728 and their LCM is 144. How many such pairs are possible?

  1. 1

  2. 3

  3. 2

  4. 5
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C Correct answer
Explanation

The product of two numbers is equal to the product of their HCF and LCM. Thus, HCF * 144 = 1728, so HCF = 12. Let the numbers be 12a and 12b where a and b are coprime; then 12a * 12b = 1728 implies ab = 12. The coprime pairs (a, b) are (1, 12) and (3, 4), yielding two pairs of numbers.

AI explanation

Using the identity Product = Highest Common Factor * Least Common Multiple, the highest common factor of the two numbers is 1728 / 144, which equals 12. Let the numbers be 12a and 12b, where a and b are coprime integers, making their product 1728 equal to 144ab; thus, ab = 12. The only pairs of coprime positive integers whose product is 12 are (1, 12) and (3, 4), meaning exactly 2 such pairs are possible.

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