CAT 2024

Slot 1

QA

Question & Solution

To find the value of $10^{100}mod\left(7\right)$
When 10 is divided by 7, it leaves a remainder 3, so the above equation can be written as, 
$3^{100}mod\left(7\right)$

Now looking at the cyclicality of powers of 3 when divided by 7, 
$3^1mod  7=3$

$3^2mod  7=2$

$3^3mod  7=6$

$3^4mod  7=4$

$3^5mod  7=5$

$3^6mod  7=1$

From this calculation, it is evident that the powers of 3 modulo 7 repeat every 6 steps. This forms a cycle: 3, 2, 6, 4, 5, 1

$3^{100}=\left(3^6\right)^{16}\times\ \left(3^4\right)$

Since $3^6mod 7=1$

We just need to consider $3^4mod 7$ which equals 4

Hence the answer is 4.