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What are the final two digits of 7^177

  • math -

    Phi(100) = 40, so the order of 7 (i.e. the smallest power to which you have to raise 7 to get 1) is a divisor of 40, and you easily find that it is 4.

    Computing Mod 100 in the following:

    7^4 = 2401 = 1

    176 is a multiple of 4, so we have:

    7^177 = 7

  • math -

    Hint: take a look at the last two digits for the first few powers of 7

    7^1 = 7 or 07
    7^2 = 49
    7^3 = 343
    7^4 = 2401
    7^5 = 16807
    7^6 = ......49
    7^7 = ......43
    7^8 = ......01
    7^9 = ......07
    7^10 = .... 49

    I am sure you can do something with this pattern.

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