mikemcgarry wrote:
guerrero25 wrote:
What is the remainder when \([(((((((43717)^4)^3)^6)^2)^8)^2)^3]^2\) is divided by 5
(A)1
(B)2
(C)3
(D)4
(3)5
Dear
guerrero25First of all, I believe all the parentheses I added are absolutely necessary to make clear what the question is asking. Remember that parentheses are not mathematical garnish, like parsley served with a meal. They have a crucial function in many problems.
The problem is way-over-the-top harder than anything the GMAT would ask. Having said that, the principle is something the GMAT does test. The principle is: in large powers, the units digit of the power depends only on the units digit of the base. More generally, if we multiply A x B = C, the units digit of C is determined exclusively by the units digits of A & B --- none of the other digits of A & B have any influence on the units digit of C.
First of all, we can totally ignore the first four digits of the base, 43717 ---- only the 7 at the end matters.
What is the units digit of 7^4?
7x7 = 49, so that's a units digit of 9.
9 x 7 = 63, so (7^3) has a units digit of 3
3 x 7 = 21, so (7^4) has a units digit of 1
Well, this is an incredible stroke of luck, because we now have a units digit of 1, and all subsequent powers of this will be 1, because 1 to any power is simply 1. Therefore, the final power, some god-awful number, must have a units digit of 1, and when divided by 5, it has a remainder of 1. Answer =
(A).
BTW, just out of curiosity, I checked this number on Wolfram Alpha. The power stated in the prompt, if fully calculated out, would have 64,153 decimal pages. In typical fonts (around 3000 characters per page), it would take 21 full pages and would spill onto the 22nd page to print this number out. A large-ish number!
Mike
Original question reads:
What is the remainder when 43717^(43628232) is divided by 5?The remainder when 43717^(43628232) is divide by 5 will be the same as the remainder when 7^(43628232) is divided by 5 (we need only the units digit to get the remainder upon division by 5).
7^1=7 divided by 5 yields the remainder of 2;
7^2=49 divided by 5 yields the remainder of 4;
7^3=343 divided by 5 yields the remainder of 3;
7^4=...1 divided by 5 yields the remainder of 1.
7^5=...7 divided by 5 yields the remainder of 2 AGAIN.
The remainders repeat in blocks of four {2, 4, 3, 1}, {2, 4, 3, 1}, ...
43628232 (exponent) is divisible by 4 (a number is divisible by 4 if its last 2 digits (32 in our case) divisible by 4). Therefore, the remainder when 43717^(43628232) is divided by 5 is the fourth number in pattern, which is 1.
Answer: A.
Hope it's clear.