Discussion
The sum and HCF of two numbers are 156 and 13. The numbers of such number pairs is
Answer: A
If the HCF is 13, both numbers are 13 times something, and those somethings share no factor. Write the pair as \( 13a \) and \( 13b \) with \( \gcd(a,b)=1 \):
\[ 13a + 13b = 156 \quad\Longrightarrow\quad a + b = 12 \]
So the question becomes: how many coprime pairs sum to 12?
- \( (1,11) \) - coprime ✓ giving 13 and 143
- \( (5,7) \) - coprime ✓ giving 65 and 91
- \( (2,10),\,(3,9),\,(4,8),\,(6,6) \) - each shares a factor, so the HCF of the pair would exceed 13
Two pairs qualify. The coprimality condition is the whole point: without it the HCF would be larger than 13 and the given data would be contradicted.
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