Find the number of pairs of positive integers (m,n) with `m le n`, such that the 'least common multiple' (LCM) of m and n equals 600.

Answer

To find the number of pairs of positive integers \((m, n)\) such that \(m \leq n\) and the least common multiple (LCM) of \(m\) and \(n\) equals 600, we can follow these steps:

### Step 1: Factorize 600
First, we need to factorize the number 600 into its prime factors.
\[ 600 = 2^3 \times 3^1 \times 5^2 \]

### Step 2: Express \(m\) and \(n\) in terms of their prime factors
Let:
\[ m = 2^{\alpha_2} \times 3^{\alpha_3} \times 5^{\alpha_5} \]
\[ n = 2^{\beta_2} \times 3^{\beta_3} \times 5^{\beta_5} \]
where \(\alpha_i\) and \(\beta_i\) are non-negative integers representing the powers of the respective prime factors.

### Step 3: Set conditions for LCM
The condition for the LCM of \(m\) and \(n\) to equal 600 can be expressed as:
\[ \text{lcm}(m, n) = 2^{\max(\alpha_2, \beta_2)} \times 3^{\max(\alpha_3, \beta_3)} \times 5^{\max(\alpha_5, \beta_5)} = 2^3 \times 3^1 \times 5^2 \]
This gives us the following inequalities:
1. \(\max(\alpha_2, \beta_2) = 3\)
2. \(\max(\alpha_3, \beta_3) = 1\)
3. \(\max(\alpha_5, \beta_5) = 2\)

### Step 4: Determine valid pairs for each prime factor
#### For \(2^{\max(\alpha_2, \beta_2)} = 2^3\):
- Possible pairs \((\alpha_2, \beta_2)\) are:
- (0, 3)
- (1, 3)
- (2, 3)
- (3, 3)
- (3, 0)
- (3, 1)
- (3, 2)

This gives us 7 valid pairs.

#### For \(3^{\max(\alpha_3, \beta_3)} = 3^1\):
- Possible pairs \((\alpha_3, \beta_3)\) are:
- (0, 1)
- (1, 1)
- (1, 0)

This gives us 3 valid pairs.

#### For \(5^{\max(\alpha_5, \beta_5)} = 5^2\):
- Possible pairs \((\alpha_5, \beta_5)\) are:
- (0, 2)
- (1, 2)
- (2, 2)
- (2, 0)
- (2, 1)

This gives us 5 valid pairs.

### Step 5: Calculate total combinations
Now, we can calculate the total number of combinations of \((m, n)\):
\[ \text{Total combinations} = 7 \times 3 \times 5 = 105 \]

### Step 6: Adjust for \(m \leq n\)
Since we are interested in pairs where \(m \leq n\), we need to consider the cases where \(m = n\) and subtract those from the total combinations. The only case where \(m = n\) is when both are equal to 600, which is one case.

Thus, the number of pairs where \(m \leq n\) is:
\[ \text{Valid pairs} = \frac{105 - 1}{2} + 1 = \frac{104}{2} + 1 = 52 + 1 = 53 \]

### Final Answer
The number of pairs of positive integers \((m, n)\) such that \(m \leq n\) and \(\text{lcm}(m, n) = 600\) is:
\[ \boxed{53} \]