# How many positive integers less than 100 are the product of three distinct primes? [ ]?

The number of positive integers less than 100 are the product of three distinct primes will be 4.

What is Algebra?

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

A prime number is a positive integer having exactly two factors that are 1 and the number itself. Any number that does not follow this is termed a composite number.

The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, ... and so on.

If the product of 2, 3, and the other prime number will be

2 x 3 x 5 = 30

2 x 3 x 7 = 42

2 x 3 x 11 = 66

2 x 3 x 13 = 78

2 x 3 x 17 = 102

If the product of 3, 5, and the other prime number will be

3 x 5 x 7 = 105

The number of positive integers less than 100 are the product of three distinct primes will be 4.

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1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

1x2x47 = 94, hence there are 14 combos with 1x2xn.
1x3x31 = 93, therefore there are only 9 distinct prime combinations here.
1x5x19 = 95, ergo there are only 5 qualifying combos here.
1x7x13 = 91, so here's 2 more possible combinations.

So there are only 30 combinations using the prime #1 in this particular scenario.  Now to start with the only even prime #, #2 to find more combos that suffice these criteria.

2x3x13 is the last one since 2x3x17 = 102.
Therefore there are only 4 possible combos here.
2x5x7 is the only combo that works for the 2x5xn model, so add 1 more.
That is the last 2xnxn model that will work.  Now for 3xnxn

3x5x7 yields 105, therefore, it is logically and theoretically impossible for another nxnxn combo to exist ≤ 100.

So to recap this:
1xnxn = 30 combos
2xnxn = 5 combos
3xnxn and higher had 0 combos (for increasing n values).

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