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 co-prime numbers. 
We are given that \( m \) and \( n \) are two-digit numbers. 
Thus: \[ 10 \leq m \leq 99 \quad \text{and} \quad 10 \leq n \leq 99 \] So: \[ 10 \leq 6a \leq 99 \quad \Rightarrow \quad 2 \leq a \leq 16 \] and \[ 10 \leq 6b \leq 99 \quad \Rightarrow \quad 2 \leq b \leq 16 \] 
Thus, \( a \) and \( b \) are integers, and the pairs \( (a, b) \) where \( \gcd(a, b) = 1 \) and \( a<b \) are the valid solutions. 
Now, consider the valid values of \( a \) and \( b \), where both are between 2 and 16 and co-prime. 
The valid pairs are as follows: 
- \( a = 2, b = 3, 5, 7, 9, 11, 13, 15 \) 
- \( a = 3, b = 4, 5, 7, 8, 10, 11, 13, 14, 16 \) 
- \( a = 4, b = 5, 7, 9, 11, 13, 14, 16 \) 
- \( a = 5, b = 6, 7, 8, 9, 11, 13, 14, 15 \) 
- \( a = 6, b = 7, 9, 11, 13, 15 \) 
- \( a = 7, b = 8, 9, 10, 11, 13, 14, 16 \) 
- \( a = 8, b = 9, 11, 13, 15 \) 
- \( a = 9, b = 10, 11, 13, 14, 16 \) 
- \( a = 10, b = 11, 13, 15 \) 
- \( a = 11, b = 12, 13, 14, 15 \) 
- \( a = 12, b = 13, 14, 15, 16 \) 
- \( a = 13, b = 14, 15, 16 \) 
- \( a = 14, b = 15, 16 \) 
- \( a = 15, b = 16 \) 
Thus, there are 64 such ordered pairs. 
Therefore, the correct answer is \( 64 \).

Answer 2

Step 1: Let \( m = 6a \) and \( n = 6b \), where \( a \) and \( b \) are co-prime numbers. 

Step 2: We are given that:

\[ m < n \quad \Rightarrow \quad a < b \]

Step 3: Since \( m \) and \( n \) are two-digit numbers, we have:

\[ 10 \leq m \leq 99 \quad \text{and} \quad 10 \leq n \leq 99 \] This implies: \[ 2 \leq a \leq 16 \quad \text{and} \quad 2 \leq b \leq 16 \]

Step 4: Now, since \( a < b \) and \( a \) and \( b \) are co-prime, we consider the following pairs of \( a \) and \( b \) that satisfy these conditions:

For \( a = 2 \), \( b = 3, 5, 7, 9, 11, 13, 15 \)

For \( a = 3 \), \( b = 4, 5, 7, 8, 10, 11, 13, 14, 16 \)

For \( a = 4 \), \( b = 5, 7, 9, 11, 13, 15 \)

For \( a = 5 \), \( b = 6, 7, 8, 9, 11, 12, 13, 15, 16 \)

For \( a = 6 \), \( b = 7, 11, 13 \)

For \( a = 7 \), \( b = 8, 9, 10, 11, 12, 13, 15, 16 \)

For \( a = 8 \), \( b = 9, 11, 13, 15 \)

For \( a = 9 \), \( b = 10, 11, 13, 14, 16 \)

For \( a = 10 \), \( b = 11, 13, 14, 16 \)

For \( a = 11 \), \( b = 12, 13, 14, 15, 16 \)

For \( a = 12 \), \( b = 13, 14, 15, 16 \)

For \( a = 13 \), \( b = 14, 15, 16 \)

For \( a = 14 \), \( b = 15, 16 \)

For \( a = 15 \), \( b = 16 \)

Step 5: Total number of ordered pairs is \( 64 \).