GATE 2019

Question

SOLVE: 2^32 mod 5

Comments

The question was 3^51 mod 5

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Thanks for correction. I haven't given that exam though.

Answer 1

I don't think specifically these were the numbers.

Still to answer this question, 2 is co-prime to 5 so using Fermat's Little Theorem (or directly), $2^4 \equiv 1 \;(\text{mod } 5)$.

$$\begin{align}
2^{32}
&\equiv (2^4)^8 \quad (\text{mod } 5) \\
&\equiv 1^8 \quad (\text{mod } 5) \\
&\equiv 1 \quad (\text{mod } 5)
\end{align}$$

Comments

Also there is a pattern 2,4,8,6,2,4,8,6,..... so whenever power is divisible by 4 you get 6 in the unit place, on divided by 5 you will get 1 as a reminder.

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@ Swapnil Naik  But if the question was 2^32 mod 4 , then 6%4=2 (As unit digit is 6)  but 16%4=0. (Unit digit is 6 again ) So multiple results would be possible according to your method.

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In this case it was mod 5 and in pattern there was no way 5 could divide others. Here to check mod we need to consider last 2 digits and in the case of 5 there was no way we could actually get a number which is divisible because of pattern. Hence I thought pattern is useful here.