Calculator will blow off (1)

Number Theory Level 4

$\large \displaystyle 1^1 \times 2^2 \times 3^3 \times\cdots \times 23^{23} \times 24^{24} \times 25^{25}$

$\,$

Find the last three non-zero digits of the number above.

Try Problem 2 and Problem 3 after solving this.

1 solution

Zyberg Nee
Jun 25, 2016

@Samara Simha Reddy , could you please post a firm solution to the problem?

The way I did it:

Finding all 5 factors, we get 5^100. So, there are going to be 100 zeros. What we need to find is:

$\frac{1^1 \times 2^2 \times 3^3 \times\cdots \times 23^{23} \times 24^{24} \times 25^{25}}{10^{100}} \pmod{1000}$

By using Chinese remainder theorem we can see that in mod 8 the product is definitely 0.

What I did with 125 was brute forcing to the end to get 824.

It's not really a good solution, but after spending a few hours on it, I couldn't think on anything smarter.