Let m and n, (m

Question

Let $m$ and $n$, ($m<n$), be two 2-digit numbers. Then the total number of pairs $(m,n)$ such that $\gcd(m,n)=6$, is \_\_\_\_\_\_\_\_\_\_.

Answer: 64

Accepted Answer

Write $m=6a,\ n=6b$ with $\gcd(a,b)=1$. Since $m,n$ are 2-digit numbers ($10\le m,n\le99$): $a,b \in \{2,3,\ldots,16\}$.

We need coprime pairs $(a,b)$ with $a<b$ in $\{2,\ldots,16\}$. Counting for each $a$:

$$a=2:7,\ a=3:9,\ a=4:6,\ a=5:9,\ a=6:3,\ a=7:8,\ a=8:4,\ a=9:5,$$
$$a=10:2,\ a=11:5,\ a=12:1,\ a=13:3,\ a=14:1,\ a=15:1$$

$$\text{Total} = 7+9+6+9+3+8+4+5+2+5+1+3+1+1 = 64$$

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