# Math Help - Exponential

1. ## Exponential

Question: What digit is found in the units place of 7 to the 1000th power ?

Can 7^1000 be broken down to 7^100 + 7^10?
3.23447651 × 10^84 + 282 475 249
so would the answer be 2?

2. Woah

Une française

Can 7^1000 be broken down to 7^100 + 7^10?
No, you can't !

$a^{bc} = (a^b)^c$

Use the congruences (you learn it in last year of high school). As you want the digit in the units, study the congruence of 7^1000 to 10.

This is why you'll try the very first powers of 7 and see what is their congruence to 10.

$7 \equiv -3 [10]$

$7^2 \equiv 9[10] \equiv -1 [10]$

So $7^4 \equiv 1 [10]$

Hence for all $k \in \mathbb{Z}$, 7^{4k} \equiv 1 [10][/tex]

-> as 1000 is 4*250, $7^{1000} \equiv 1 [10]$

So the digit unit will be 1.