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# What is the remainder when 43^43+ 33^33 is divided by 10

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CEO
Joined: 21 Jan 2007
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What is the remainder when 43^43+ 33^33 is divided by 10 [#permalink]

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04 Jan 2008, 14:17
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

What is the remainder when 43^43+ 33^33 is divided by 10?
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You tried your best and you failed miserably. The lesson is 'never try'. -Homer Simpson

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Director
Joined: 08 Jun 2007
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Re: exponents - remainders [#permalink]

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04 Jan 2008, 14:37
(43^4)^10 * 43^3 + (33^4)^8 * 33

unit digit 1 * unit digit 7 + unit digit 1 * unit digit 3 = unit digit 0

So divisible by 10.

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VP
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Re: exponents - remainders [#permalink]

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04 Jan 2008, 16:16
bmwhype2 wrote:
What is the remainder when 43^43+ 33^33 is divided by 10?

3^1=3
3^2=9
3^3=27
3^4=81
3^5=243
...........
3^33=...3
3^43=...7

remainder is 0

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Manager
Joined: 04 Jan 2008
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Re: exponents - remainders [#permalink]

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04 Jan 2008, 19:59
remainder 0
good qn

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CEO
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Re: exponents - remainders [#permalink]

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05 Jan 2008, 06:20
ashkrs wrote:
(43^4)^10 * 43^3 + (33^4)^8 * 33

unit digit 1 * unit digit 7 + unit digit 1 * unit digit 3 = unit digit 0

So divisible by 10.

what is the purpose of breaking it down like you did? i don't quite follow
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Director
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Re: exponents - remainders [#permalink]

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05 Jan 2008, 06:31
bmwhype2 wrote:
What is the remainder when 43^43+ 33^33 is divided by 10?

All we care about is the units digit of 43^43 and the units digit of 33^33. Both are dependent on powers of 3.
Units digit of powers of 3 are:
3^1 = 3
3^2 = 9
3^3 = 7
3^4 = 1

3^43 = (3^4)^10 * 3^3 = 1*7 = 7
3^33 = (3^4)^7 * 3^5 = 1*3 = 3

Remainder of (7+3)/10 is 0. That's the answer.

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CEO
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Re: exponents - remainders [#permalink]

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05 Jan 2008, 21:30
why do we only look at the units digit?
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Senior Manager
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Re: exponents - remainders [#permalink]

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06 Jan 2008, 00:16
bmwhype2 wrote:
why do we only look at the units digit?

because all multiples of 10 have their units digit as 0, and no other.
so once we prove that the units digit is 0,we crack the ans.
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Re: exponents - remainders   [#permalink] 06 Jan 2008, 00:16
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# What is the remainder when 43^43+ 33^33 is divided by 10

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