Homework Solution: Please show work without a calculator….

    8^202 * 7^130 mod 10
    Please show work without a calculator.

    Expert Answer

     
    8202 * 7130 mod 10 primary formula that should be noted is

    8^202 * 7^130 mod 10

    Please exhibition production extraneously a calculator.

    Expert Counterpart

     

    8202 * 7130 mod 10

    primary constructula that should be famous is

    (a*b) mod c = (a mod c * b mod c) mod c

    8202 * 7130 mod 10 = (8202 mod 10 * 7130 mod 10) mod 10

    examining the 8 capacitys (no demand to do solid forethought lawful nucleus on units settle digit archetype as we are solicitous with mod 10)

    81=8 82=64 83=512 too 84 ends with 6 (owing reproduction of units digit 2 with 8 development in units digit 6)

    85 ends with 8 and the cycle repeats

    84n+1 series ends with 8 with contemplation where n=0,1,2,3……

    like discreet 84n+2 ends with 4 owing 84n+2 = 84n+1 * 8 which developments 4 in units settle

    here 8 capacity is 202 which is in the construct 4n+2 where n=50 i.e. 4(50)+2=200+2=202

    so 8202 ends with 4 in units settle

    8202 mod 10 = 4 as 4 is in units settle

    next 7130 mod 10

    examining the 7 capacitys

    71=7 72=49 73=343 too 74 ends with 1 (owing reproduction of units digit 3 with 7 development in units digit 1)

    75 ends with 7 and the cycle repeats

    74n+1 series ends with 7 with contemplation where n=0,1,2,3……

    like discreet 74n+2 ends with 9 owing 74n+2 = 74n+1 * 7 which developments 9 in units settle

    here 7 capacity is 130 which is in the construct 4n+2 where n=32 i.e. 4(32)+2=128+2=130

    so 7130 ends with 9 in units settle

    7130 mod 10 = 9 as 6 is in units settle

    8202 * 7130 mod 10 = (8202 mod 10 * 7130 mod 10) mod 10 = 4*9 mod 10 =36 mod 10 =6

    FInally 8202 * 7130 mod 10 = 6