Determine 7^1000 mod 6 and 6^1001 mod 7.?

NetherCraft 0

Please show me how you did it. Thanks for your help.

4 Answers

  • If you think of seven to its early powers you will see the mod: (remainder when divided by 6)

    7/6 has R:1

    49/6 has R:1

    343/6 has R:1 and

    2401/6 has R:1

    7 to any power in mod 6 has a remainder of 1

    Since 7 is 6 with a remainder of 1 we can treat 7^1000 mod 6 as 1^1000 mod 6 which is 1 divided by 6 which of course has remainder 1.

    Similarly for 6^1001 mod 7, if we divide the early powers of 6 by 7 we see the pattern in the remainders.

    6/7 has remainder 6

    36/7 has remainder 1

    216/7 has remainder 6

    1296/7 has remainder 1

    We see that 6 to an odd power divided by 7 has remainder 6

    and 6 to an even power divided by 7 has remainder 1

    so 6^1001 mod 7 being an odd power will have a remainder of 6.

    We could say that 6 in mod 7 is -1

    so -1^1001 mod7 = -1 mod 7….and since 7-1=6 the answer is 6.

    In short

    7^1000 mod 6 = 1^1000 mod 6 = 1 mod6 = 1

    6^1001 mod 7 = (-1)^1001 mod 7 = -1 mod 7 = 7-1 = 6

  • The remainder of the products equals the remainder of the product of the remainders.

    Examples

    6 mod 5 = 1

    8 mod 5 = 3

    (68) mod 5 = (13) mod 5 = 3

    11 mod 7 = 4

    1111 mod 7 = (44) mod 7 = 2

    Now lets try the first one:

    7 mod 6 = 1

    7^1000 mod 6 = 1^1000 mod 6 = 1

    The second one is more difficult:

    6 mod 7 = 6

    The above law results in the same expression.

    6^1001 = 6^1001 mod 6

    Let’s write 6^1001 a different way:

    6^1001 = (6^1000)6 = (36^500)6 mod 6

    36 mod 7 = 1

    36^500 mod 7 = 1^500 mod 7 = 1

    (36^500)6 = (16) mod 7 = 6

  • 7^1000 mod 6 = 1^1000 mod 6 = 1.

    6^1001 mod 7 =(-1)^1001 mod 7 = -1 = 6.

  • 7^1000 mod 6 is 7 mod 6 done a 1000 times or 1 * 1000 =1

    and so forth.

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