Question

The HCF of 2472, 1284 and a third number 'N' is 12. If their LCM is $2^3\times 3^2\times 5^1\times 103\times 107$, then the number 'N' could be:

• A. $2^2\times 3^2\times 7^1$
• B. $2^2\times 3^3\times 103$
• C. $2^2\times 3^2\times 5^1$
• D. None of these

Answer 1

The correct option is C $2^2\times 3^2\times 5^1$

$2472=2^3\times 103\times 3;1284=2^2\times 107\times 3$. Since the HCF is 12, the number must have a component of $2^2\times 3$ at the very least in it. Also, since the LCM is $2^3\times 3^2\times 5^1\times 103\times 107$. We can see that the minimum requirement in the required number has to be $3^2\times 5^1$. Combining these two requirements we get that the number should have $2^2\times 3^2\times 5^1$ at the minimum and the power of 2 could also be $2^3$ while we could also have either one of 103 and/or 107 as a part of the required number.
Thus, for instance, the number could also be $2^3\times 3^2\times 5^1\times {103}^1\times {107}^1$. The question has asked us what 'could' the number be?
Option (c) gives us a possible value of the number and is hence the correct answer.