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what is the unit digit of 3^65 X 6^59 X 7^71?

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by Anurag@Gurome » Mon Jun 18, 2012 12:00 am
Chinn_asama wrote:What is the unit digit of 3^65 X 6^59 X 7^71?
Unit's digit of (3^65)*(6^59)*(7^71) = (Unit's digit of 3^65)*(Unit's digit of 6^59)*(Unit's digit of 7^71)

Any power of 6 ends with 6.
Hence, unit's digit 6^59 = 6

Now for powers of 3,
  • Unit's digit of 3^(Multiple of 4) = 1
    Unit's digit of 3^(Multiple of 4 + 1) = 3
    Unit's digit of 3^(Multiple of 4 + 2) = 9
    Unit's digit of 3^(Multiple of 4 + 3) = 7
Hence, unit's digit of 3^65 = unit's digit of 3^(4*16 + 1) = 3

Now for powers of 7,
  • Unit's digit of 7^(Multiple of 4) = 1
    Unit's digit of 7^(Multiple of 4 + 1) = 7
    Unit's digit of 7^(Multiple of 4 + 2) = 9
    Unit's digit of 7^(Multiple of 4 + 3) = 3
Hence, unit's digit of 7^71 = unit's digit of 7^(4*17 + 3) = 3

Hence, unit's digit of (3^65)*(6^59)*(7^71) = unit's digit of 3*6*3 = unit's digit of 54 = 4
Chinn_asama wrote:what is the unit digit of 7^105
Unit's digit of 7^105 = Unit's digit of 7^(4*26 + 1) = 7
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by Chinn_asama » Mon Jun 18, 2012 1:31 am
But how do you that for a different number, say 8/12. Is there some logic? please explain

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by dhonu121 » Mon Jun 18, 2012 4:25 am
Boils down to 3*6*3 = 4.
Thus units digit is 4.
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by Anurag@Gurome » Mon Jun 18, 2012 4:31 am
Chinn_asama wrote:But how do you that for a different number, say 8/12.
For powers of 8, (powers being positive)
  • Unit's digit of 8^(Multiple of 4) = 6
    Unit's digit of 8^(Multiple of 4 + 1) = 8
    Unit's digit of 8^(Multiple of 4 + 2) = 4
    Unit's digit of 8^(Multiple of 4 + 3) = 2
For powers of 12, (powers being positive)
  • Unit's digit of 12^(Multiple of 4) = Unit's digit of 2^(Multiple of 4) = 6
    Unit's digit of 12^(Multiple of 4 + 1) = Unit's digit of 2^(Multiple of 4 + 1) = 2
    Unit's digit of 12^(Multiple of 4 + 2) = Unit's digit of 2^(Multiple of 4 + 2) = 4
    Unit's digit of 12^(Multiple of 4 + 3) = Unit's digit of 2^(Multiple of 4 + 3) = 8
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