Big 30

Number Theory Level 4

Suppose 30 = a b c , 30 = abc , where a , a , b , b , and c c are all positive integers.

How many ordered triples of solutions ( a , b , c ) (a, b, c) are there?


The answer is 27.

Jason Dyer by Jason Dyer

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1 solution

Sharky Kesa
Jan 10, 2017

30 = 2 × 3 × 5 30=2\times 3 \times 5

Each of these prime factors are assigned to one of the three variables. Since there are 3 distinct primes, that means there are 3 3 = 27 3^3=27 different ordered ways to do this.

8 Helpful 1 Interesting 0 Brilliant 0 Confused

Why Not 6?
3! = 3 * 2 * 1
You cannot use the one prime multiple Times. 2 * 2 * 5 is not 30 and so on...

Eric Scholz - 2 years ago

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Same problem here, typed 6. Triples of solutions (a, b, c) means an ordered triplet that satisfies 30 = abc, so 27 is clearly wrong.

A Former Brilliant Member - 1 year, 12 months ago

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Tuples are always assumed to be ordered. Ordered triple means that the order matters i.e. ( a , b , c ) (a, b, c) is distinct to ( a , c , b ) (a, c, b) . Note that the empty product is 1, so if you choose 0 primes, then your number is 1. Thus, 27 is correct.

Sharky Kesa - 1 year, 12 months ago

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