Remainder questions

For Q1:

80/4 = Remainder 0
So, (2)^4 = 16
16/24 --> remainder 16


ANSWER {B}

For Q2:

190/4 = Remainder 2

2^2= 4

4/1729 --> remainder 4

Are you sure that answer is not {E}??

theCodeToGMAT wrote:For Q2:

190/4 = Remainder 2

2^2= 4

4/1729 --> remainder 4

Are you sure that answer is not {E}??

Yes, I am sure answer is not E.

vinni.k wrote:

theCodeToGMAT wrote:For Q2:

190/4 = Remainder 2

2^2= 4

4/1729 --> remainder 4

Are you sure that answer is not {E}??

Yes, I am sure answer is not E.

Oh, do you have the explanation of the Solution?

I googled it and find out this is beyond the scope og GMAT, here is the full explaination by Expert Mike

Notice that 12^3 = 1728, so this divisor is 1729 = ((12^3) + 1). We will use that to our advantage.

12^190 = (12^3)*(12^187) = (12^3)*(12^187) + (12^187) - (12^187)
12^190 = [(12^3)+1]*(12^187) - (12^187)
12^190 = [(12^3)+1]*(12^187) - (12^3)*(12^184)
12^190 = [(12^3)+1]*(12^187) - (12^3)*(12^184) - (12^184) + (12^184)
12^190 = [(12^3)+1]*(12^187) - [(12^3)+1]*(12^184) + (12^184)
12^190 = (1729)*(12^187) - (1729)*(12^184) + (12^184)
The two purple terms are divisible by 1729, so when divided by 1729, they will have a remainder of zero. The green term, when divided by 1729, will have the same remainder as does 12^190 when divided by 1729. That's interesting --- we can use this trick to create a smaller number with the same remainder.

Notice, we could repeat this trick, and bring the number down by a factor of 12^6 again and again. The number 180 is certainly divisible by 6, so 186 must be----- we could drop the power of 12 from 12^190 all the way down to 12^4, that is, 186 powers lower, and we would still have the same remainder when divided by 1729. So, now the whole problem reduces to --- what is the remainder when 12^4 is divided by 1729?

12^4 = (12^3)*(12) = (12^3)*(12) + 12 - 12 = [(12^3)+1]*(12) - 12

So, when 12^4 is divided by 1729, we get the same remainder as when -12 is divided by 1729. OK, that's a little confusing, to have a negative dividend, but when we have one number with a certain remainder, all we have to do is add or subtract the divisor (or a multiple of the divisor) to get other numbers with t the same remainder. Here, I will just add 1729

(-12) + 1729 = 1717

Of course, 1717 < 1729, so when 1717 is divided by 1729, 1729 goes into it zero times with a remainder of 1717. That's the answer.

Does all this make sense?

Mike
_________________

Mike McGarry
Magoosh Test Prep

vinni.k wrote:Find the remainder when 2^80 is divided by 24 ?

A). 8
B). 16
C). 4
D). 2
E). 6

OA is B

24 = 8*3
2^80 = (8*3)Q + R
2^77 = 3Q + (R/8)
The remainder when 2^77 is divided by 3 is (-1)^77 = -1 or 2
R/8 = 2
R = 16

Choose B

vinni.k wrote:What is the remainder if 12^190 is divided by 1729 ?

A. 12
B. 1
C. 1728
D. 1717
E. 4

OA is D

Please explain

Thanks & Regards
Vinni

1729 = 12^3 + 1
12^190 = (12^3)^63 * 12
Remainder when 12^3 is divided by 1729 = -1
Remainder when (12^3)^63 is divided by 1729 = (-1)^63 = -1
Remainder when (12^3)^63 * 12 is divided by 1729 = -1 * 12 = -12 or (1729-12) = 1717

Choose D

Cheers