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?

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- Mar 20th 2008, 12:27 AMla_bordéliqueExponential
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? - Mar 20th 2008, 12:39 AMMoo
Woah

Une française :D

Quote:

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.