How many could it be?

Number Theory Level pending

How many integers less than 1000 has exactly three (3) factors?


The answer is 11.

Juvin Abayon by Juvin Abayon , 24, Philippines

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1 solution

Stephen Mellor
Jan 5, 2018

Odd number of factors → Square number.

Only 3 factors → Square of a prime.

Primes where their square is less than 1000: 2,3,5,7,11,13,17,19,23,29,31.

Hence there are 11 integers.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It is true that a square of a prime has only three factors (1, p, and p^2, and no others as p is prime), but a perfect square number doesn't always have an odd number of factors, e.g., 81, which has four distinct factors, 1, 3, 9 and 81.

Juvin Abayon - 3 years, 5 months ago

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Squares always have odd number of factors as they pair up with others, apart from the square root which pairs up with itself. In your example, what pairs up with 3? Perhaps 27? ;)

Stephen Mellor - 3 years, 5 months ago

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Coming to think of it... I see, I stand corrected my good sir, thanks for pointing that out. :D

Juvin Abayon - 3 years, 5 months ago

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