Factorial of Smallest Multiple!

Number Theory Level 3

Find the smallest positive integer value of $x$ such that $(349x)!$ is divisible by $349^{350}$ .

If there is no such positive integer value $x$ , then enter 0.

Is constructing a generalization possible after replacing $349$ by any prime $p$ and $350$ by $p+1$ ? If No , why?

The answer is 349.

1 solution

Kushal Bose
Nov 29, 2016

Here $349$ will be the answer because in each multiple of 349 there is only one factor of 349 available.In the last rem $349 \times 349$ we get one more extra $349$ .So,total become $349+1=350$ .

Yes it is possible to generalize for any prime $p$ because no number will prime number untill it is the factor of that prime

Factorial of Smallest Multiple!

The answer is 349.

1 solution

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