Remainder?

**a69356** · October 26th 2009, 02:31 AM

Q) Find the remainder of 37^47^57 divided by 16?

Any help will be appreciated?

Thanks,
Ashish

**Soroban** · October 26th 2009, 10:49 AM

Hello, Ashsh!

I think I've solved it . . .

Find the remainder of $37^{47^{57}}$ divided by 16

Since $37 \equiv 5 \text{ (mod 16)}$

. . we have: . $5^{47^{57}} \text{ (mod 16)}$

Now consider consecutive powers-of-5 (mod 16):

. . $\begin{array}{|c|c|}<br /> 2^n & \text{(mod 16)} \\ \hline \\[-4mm]<br /> 5^1 & 5 \\ 5^2 & 9 \\ 5^3 & 13 \\ 5^4 & 1 \\ 5^5 & 5 \\ \vdots & \vdots \end{array}$

The remainders step through a 4-step cycle: . $5-9-13-1- \hdots$

Now we must determine: . $47^{57}\text{ (mod 4)}$

We have: . $47^{57}\text{ (mod 4)} \;=\;(\text{-}1)^{57}\text{ (mod 4)} \;=\;-1\text{ (mod 4)}\;=\;3\text{ (mod 4)}$

Hence: . $5^{47^{57}} \;=\;5^3\text{ (mod 4)}$

And: . $5^3 \div 16$ has remainder 13.

Therefore: . $37^{47^{57}} \div 16\,\text{ has remainder }13.$

But someone check my reasoning and work ... please!
.

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