Question

The sum of two numbers is 1056 and their HCF is 66. Find the number of such pairs.

• A. 6
• B. 2
• C. 4
• D. 8

Hint:

Use HCF to factor out common divisors, then check for co-prime conditions and valid pair counts.

Answer 1

The Correct Option is C

Let the numbers be $66a$ and $66b$, since their HCF is 66.
Then: $66a + 66b = 1056$
$\Rightarrow 66(a + b) = 1056$
$\Rightarrow a + b = \frac{1056}{66} = 16$
Now, we need to count the number of pairs of $(a, b)$ such that:
- $a + b = 16$
- $\gcd(a, b) = 1$ (because HCF of original numbers is 66)
Coprime pairs adding to 16 are:
$(1,15), (3,13), (5,11), (7,9)$ — total 4 pairs.