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 _______

Hint:

When dealing with co-prime numbers, use the properties of the greatest common divisor and the restrictions on the values to count the valid pairs.

Answer 1

Correct Answer: 64

Let \( m = 6a \) and \( n = 6b \), where \( a \) and \( b \) are coprime integers.
Given that \( m \) and \( n \) are two-digit numbers, we have \( 10 \leq m \leq 99 \) and \( 10 \leq n \leq 99 \). 
This implies \( 10 \leq 6a \leq 99 \), so \( 2 \leq a \leq 16 \), and \( 10 \leq 6b \leq 99 \), so \( 2 \leq b \leq 16 \). 
The valid solutions are pairs \( (a, b) \) such that \( 2 \leq a<b \leq 16 \) and \( \gcd(a, b) = 1 \). 
The following pairs \( (a, b) \) satisfy these conditions: 
- If \( a = 2 \), \( b \in \{3, 5, 7, 9, 11, 13, 15\} \) 
- If \( a = 3 \), \( b \in \{4, 5, 7, 8, 10, 11, 13, 14, 16\} \) 
- If \( a = 4 \), \( b \in \{5, 7, 9, 11, 13, 14, 16\} \) 
- If \( a = 5 \), \( b \in \{6, 7, 8, 9, 11, 13, 14, 15\} \) 
- If \( a = 6 \), \( b \in \{7, 9, 11, 13, 15\} \) 
- If \( a = 7 \), \( b \in \{8, 9, 10, 11, 13, 14, 16\} \) 
- If \( a = 8 \), \( b \in \{9, 11, 13, 15\} \) 
- If \( a = 9 \), \( b \in \{10, 11, 13, 14, 16\} \) 
- If \( a = 10 \), \( b \in \{11, 13, 15\} \) 
- If \( a = 11 \), \( b \in \{12, 13, 14, 15\} \) 
- If \( a = 12 \), \( b \in \{13, 14, 15, 16\} \) 
- If \( a = 13 \), \( b \in \{14, 15, 16\} \) 
- If \( a = 14 \), \( b \in \{15, 16\} \) 
- If \( a = 15 \), \( b = 16 \) 
There are a total of 64 such ordered pairs. 
Therefore, the answer is \( 64 \).