What is the remainder when 8^643/132?

What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?

Hi g3kr,

Both of the last 2 solutions fetching 29 and 116 are correct processwise, but for a logical mistake in former.

Consider the case, 16/10 will have a remainder of 6, but its not same as 8/5 which has a remainder of 3 only.

The difference? notice that for any y/x remainder can not be greater than x, hence if you have factored out a number from numerator and denominator, you have reduced remainder by that times.
In our example of 16/10: if we make it 8/5 by dividing each term by 2 and get 3 as remainder. we should multiply result by 2 to get correct ans 6.


hence, once you arrived in first solution (M33+29)

you should remember to multiply it by 4, the one which was factored out earlier. Or better dont factor out and cancel a term in a remainder problem.

Hope it helps.

What is wrong if i cancel out 2^2 in the numerator and divide by 33 . I still have to get the same reminder right?[/quote]


Hi g3kr,

Both of the last 2 solutions fetching 29 and 116 are correct processwise, but for a logical mistake in former.

Consider the case, 16/10 will have a remainder of 6, but its not same as 8/5 which has a remainder of 3 only.

The difference? notice that for any y/x remainder can not be greater than x, hence if you have factored out a number from numerator and denominator, you have reduced remainder by that times.
In our example of 16/10: if we make it 8/5 by dividing each term by 2 and get 3 as remainder. we should multiply result by 2 to get correct ans 6.


hence, once you arrived in first solution (M33+29)

you should remember to multiply it by 4, the one which was factored out earlier. Or better dont factor out and cancel a term in a remainder problem.

Hope it helps.[/quote]


Thanks,

Even I had the same ambiguity..
But just to be sure.. there wont be both options in the answer : 29 & 116 ?

Hi mindmind (too many minds :D),

Exactly what I explained with example of 16/10 and 8/5. if i ask you remainder of 16/10 and give u option of 6 and 3, will you be confused? I dont think so

If u factor out a number, the "fraction" remains same but "remainder" does not. please re-read the post. Also this is why it is important not to factor out in remainder problem.
Also try to check one more way of understanding it:
a remainder is shown as below:
if n = qm +r
then kn = kqm +kr
where Kn is a multiple of n and km is multiple of m. But this changes remainder from r to kr.

Hope it is clear..

Responding to a pm:



What is the remainder when 12 is divided by 8?
12/8 quotient 1 and remainder 4.

But when we divide both numerator and denominator by 4, we get 3/2. What is the remainder when 3 is divided by 2? It is 1. It is not the same as the remainder when 12 is divided by 8.
To get the actual remainder, you need to multiply 1 by 4 back to get 4 as remainder.

Hi, Correct me if I'm wrong.

Here, cyclicity of 8 is 4.
643/4, gives 3 as reminder.
8^3 /132 = 512/132, leaves a reminder of 116.

Different way to solve: It worked somehow and it's easier than what's being said. Find the unit's digit pattern for 8. (I've noticed all numbers' units digit pattern repeats after 4 iterations).

8^1 = 8
8^2 = 64
8^3 = 512
8^4 = 4096

Find 643/4 = 160 r3. This means that the number will end in the iteration of 8^3 = 512.

512/132 = 3 r 116­­­­