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
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Mike McGarry
Magoosh Test Prep
