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

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rashi22
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What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   04 Sep 2010, 00:15
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What is the remainder when 43^43+ 33^33 is divided by 10?

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Y is the ans 0..
i have an explanation to the question but i donot understand it please can some 1 explain


EXPLANATION:


1)Units digit for power of any number with 3 in the units place follows pattern 3,9,7,1.
2)Units digit for 43^43 = 7 (remainder of 43 divided by 3 is 1. So the units digit will be 3)
3)Units digit for 33^33 = 7
4)Units digit of (43^43 + 33^33) = Units digit for 43^33 + Units digit for 33^33 = 0 (discard the 1).
5)Units digit when it is divided by 10 is same as the units digit of the number itself = 0.

i did not understand line 4, why will we do 43^33?
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   04 Sep 2010, 06:37
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rashi22 wrote:
What is the remainder when 43^43+ 33^33 is divided by 10?

ANS: 0

Y is the ans 0..
i have an explanation to the question but i donot understand it please can some 1 explain


EXPLANATION:


1)Units digit for power of any number with 3 in the units place follows pattern 3,9,7,1.
2)Units digit for 43^43 = 7 (remainder of 43 divided by 3 is 1. So the units digit will be 3)
3)Units digit for 33^33 = 7
4)Units digit of (43^43 + 33^33) = Units digit for 43^33 + Units digit for 33^33 = 0 (discard the 1).
5)Units digit when it is divided by 10 is same as the units digit of the number itself = 0.

i did not understand line 4, why will we do 43^33?


Integer is divisible by 10 if the units digit is zero. Next, units digit of \((xyz)^n\) is the same as units digit of \(z^n\), so for example units digit of \(43^{43}\) is the same as the units digit of \(3^{43}\).

There are 2 typos in explanation:
3. Units digit of 33^33 = 3 --> units digit of \(33^{33}\) is the same as the units digit of \(3^{33}\). Cyclicity of 3 in power is 4 so \(3^{33}\) will have the same units digit as \(3^1\), so 3.

4. 43^43 + 33^33 --> some number with units digit 7 plus some number with units digit 3 = some numer with units digit 0. Number with units digit zero is divisible by 10.

Hope it helps.
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   04 Sep 2010, 06:40
last digit of 43^43 = last digit of 43^(10*4+3)= last digit of 43^3=7

last digit of 33^33= last digit of 33^(8*4+1) = last digit of 33^1= 3

please note that I have expressed the power in form of 4x+k, because power of 3 repeats after every 4 times.

So, the last digit of the sumis 7+3 = 0
therefore, the raminder will be 0.

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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   04 Sep 2010, 23:45
Understood now.. initially i was confused.. thanks
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rashi22
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   05 Sep 2010, 03:44
undersood!! thanks alot!
appreciated! :)
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   05 Sep 2010, 23:53
Thanks for the explaination Bunuel
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   17 Oct 2010, 07:53
The question says what is the remainder when divided by \(10\)?
I know that the unit digit remainder is \(0\) as sum of remainders is \(7+3=10\) but when divided by \(10\), shouldn't it be some other digit after dividing the number by \(10\) (actually \(10th\) digit remainder in this case)?

Let's suppose that last 2-digits for \(43^{43}\) are \(27\) and for \(33^{33}\) are \(33\) then the last 2-digits of the sum would be \(60\) and when divided by \(10\) it gives \(\frac{60}{10}=6\) as a remainder (not essentially \(6\), it can be any \(10th\) digit remainder).
Am I right or not?? Any expert??
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Re: What is the remainder when 43^43+ 33^33 is divided by 10? [#permalink] New post   15 Jan 2017, 14:02
HI,

Could we do the following?

last digit of 43^43 = last digit of 43^(10*4+3)= last digit of 43^3=7

last digit of 33^33= last digit of 33^(10*3+3) = last digit of 33^3= 7

If not, what are the rules to the break down the power?

Thanks
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Re: What is the remainder when 43^43+ 33^33 is divided by 10 [#permalink] New post   08 Feb 2021, 05:10
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Re: What is the remainder when 43^43+ 33^33 is divided by 10 [#permalink]
08 Feb 2021, 05:10
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