Find three pairs of numbers for which the least common multiple equals the product of the two numbers

not using one I assume :)

2,3
2,5
2,7

2 and any old prime
or any two primes like 3 and 5

To find three pairs of numbers for which the least common multiple (LCM) equals the product of the two numbers, we can start by understanding what LCM is.

The LCM of two or more numbers is the smallest number that is divisible by each of the given numbers without leaving a remainder. To find the LCM, we can use the prime factorization method or the method of finding the greatest common divisor (GCD).

Let's go through an example of finding the LCM:
Example:
Find the LCM of 4 and 6.

1. Prime factorization method:
- Prime factorization of 4: 2 * 2
- Prime factorization of 6: 2 * 3
- LCM = 2 * 2 * 3 = 12

2. Method using GCD:
- GCD of 4 and 6: 2
- LCM = (4 * 6) / GCD = (4 * 6) / 2 = 12

Now let's find three pairs of numbers for which the LCM equals the product of the two numbers.

Pair 1: (2, 2)
- LCM of 2 and 2: 2
- Product of 2 and 2: 2 * 2 = 4
- Since 2 is the LCM of 2 and 2 and matches their product, it satisfies the condition.

Pair 2: (3, 3)
- LCM of 3 and 3: 3
- Product of 3 and 3: 3 * 3 = 9
- Since 3 is the LCM of 3 and 3 and matches their product, it satisfies the condition.

Pair 3: (2, 4)
- LCM of 2 and 4: 4
- Product of 2 and 4: 2 * 4 = 8
- Since 4 is the LCM of 2 and 4 and matches their product, it satisfies the condition.

Therefore, the three pairs of numbers for which the LCM equals the product of the two numbers are (2, 2), (3, 3), and (2, 4).