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Number Theory Level 1

$\begin{aligned}a! \times b! \times c! &= 60480 \\ (a-1)!\times (b-1)!\times (c-1)! &= 1440\end{aligned}$

Given the above, what is the value of $a\times b \times c?$

The answer is 42.

2 solutions

Zach Abueg
Jul 5, 2017

$\displaystyle \begin{aligned} \frac{a!}{(a - 1)!} \times \frac{b!}{(b - 1)!} \times \frac{c!}{(c - 1)!} & = \frac{60480}{1440} \\ a \times b \times c & = \boxed{42} \end{aligned}$

I don't think that it is enough... If there exist three number such that the statement is true, then the answer is 42, but if there doesn't exist then 42 is wrong. Luckily there exist three numbers: a=2, b=3, c=7.

Áron Bán-Szabó - 3 years, 11 months ago

Then Christopher needs to specify that "there exist $3$ positive integers $(a, b, c)$ such that..."

Zach Abueg - 3 years, 11 months ago

I certainly understand what you mean, though.

Zach Abueg - 3 years, 11 months ago

Wildan Bagus Wicaksono
Jul 5, 2017

$a \times b \times c = \frac { a! }{ (a-1)! } \times \frac { b! }{ (b-1)! } \times \frac { c! }{ (c-1)! } =\frac { 60480 }{ 1440 } \\ \frac { a(a-1)! }{ (a-1)! } \times \frac { b(b-1)! }{ (b-1)! } \times \frac { c(c-1)! }{ (c-1)! } =42\\ a\times b\times c=42$

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The answer is 42.

2 solutions

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