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?
Woah
Une française
No, you can't !Can 7^1000 be broken down to 7^100 + 7^10?
$\displaystyle a^{bc} = (a^b)^c$
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.
$\displaystyle 7 \equiv -3 [10]$
$\displaystyle 7^2 \equiv 9[10] \equiv -1 [10]$
So $\displaystyle 7^4 \equiv 1 [10]$
Hence for all $\displaystyle k \in \mathbb{Z}$, 7^{4k} \equiv 1 [10][/tex]
-> as 1000 is 4*250, $\displaystyle 7^{1000} \equiv 1 [10]$
So the digit unit will be 1.