# Digit in the ten's place of an expression

What is the digit in the ten's place of $23^{41}* 25^{40}$ ? How do you calculate this? The usual method for this kind of problem is using the Binomial theorem, but I couldn't solve it.

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Note that $25^2=625\equiv 25\bmod 100$, so that in fact $25^n\equiv 25\bmod 100$ for any $n$. Because $\phi(100)=40$, by Euler's theorem we have that $a^{40}\equiv1\bmod 100$ for any $a$ relatively prime to 100 (as 23 is). Thus $23^{41}\equiv 23\bmod 100$. Now put these results together to find $$23^{41}\cdot 25^{40}\bmod 100.$$

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$$(24+1)^{40} \times (24-1)^{41}$$ $$= 23\times (24^2-1)^{40}$$ $$=23\times (575)^{40}$$

You can extract a pattern for powers of $5$. The last digits will always be $5$ and for this case $575^n$ the last two digits will alternate between $25$ (even n) and $75$ for odd n.

So $$\cdots 25$$ $$\underline{\qquad \times 23}$$ $$\quad \cdots 75$$ $$\underline{\;\cdots 50\times}$$ $$=\cdots 75$$

I have tried to emulate the multiplication procedure to show it.

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More simply, $23^{41} 25^{40} = (23 \cdot 25)^{40} 23 = 575^{40} 23$. –  Shai Covo Jun 20 '11 at 6:40
@shai you're right. When I saw the problem, somehow $24^2 =576$ came to my mind.. –  kuch nahi Jun 20 '11 at 6:48