What is the remainder if 7^10 is divided by 100?

Option (E) 49 is the correct answer.
From the repeatibility of \(7^x\) = 4, we know 7^10 will end with 9. But, since we have to check remainder when divided by 100, last two digits of the number would be our answer.

If we had only one option with 9 as the last digit, that would have been our answer, but we don't, so we need to find another way to split between 19 and 49:

7\(^4\) is 2401 . This would give remainder (1) when divided by 100.

So, we can write 7^10 as ( 7\(^4\) *7\(^4\) * 7\(^2\)) divided by 100.

This would give us (1*1*49)/ 100 which would give the remainder as 49.

Alternate:
7^10 can be written as (49)\(^5\)
The units digit of 9\(^{even}\) is 1 and the units digit of 9\(^{odd}\) is 9.

So, you can write \(\frac{(49^5)}{100}\) as \(\frac{(49^2 * 49^2 * 49)}{100}\). We know: Rem\(\frac{(49^2)}{100}\)=1 AND Rem\(\frac{(49)}{100}\)= 49.

Thus, total remainder= 1*1*49 = 49 remainder.

Note:
If the units digit of a number is 1, then the remainder when this number will be divided by 100 will have the units digit of 1, for example 231 divided by 100 gives the reminder of 31.

If the units digit of a number is 9, then the remainder when this number will be divided by 100 will have the units digit of 9, for example 239 divided by 100 gives the reminder of 39.