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Math Help - Exponential

  1. #1
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    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?
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    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.
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