What is the remainder when 10^ 49 +2 is divided by 11 ?
1
2
3
5
7
Number properties
This topic has expert replies
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Unless I'm missing a completely different solution, this question relies on the rule for determining whether a number is divisible by 11 (https://www.math.hmc.edu/funfacts/ffiles/10013.5.shtml). I've never seen an official question rely on this rule, so in my opinion, it's out of scope for the GMAT.sud21 wrote:What is the remainder when 10^ 49 +2 is divided by 11 ?
1
2
3
5
7
Cheers,
Brent
-
- Master | Next Rank: 500 Posts
- Posts: 126
- Joined: Sat Jun 07, 2014 5:26 am
- Thanked: 3 times
I think the exponent rule will have a role here.
10^1+2-Remainder is 1
10^2+2-Remainder is 3
10^3+2-Remainder is 1
10^4+2-Remainder is 3
.
.
.
So on.
SO 10^49+2 will have remainder of 1.Ans-A
10^1+2-Remainder is 1
10^2+2-Remainder is 3
10^3+2-Remainder is 1
10^4+2-Remainder is 3
.
.
.
So on.
SO 10^49+2 will have remainder of 1.Ans-A
-
- Master | Next Rank: 500 Posts
- Posts: 126
- Joined: Sat Jun 07, 2014 5:26 am
- Thanked: 3 times
If u have noticed :Here every Odd power of 10 would produce remainder of 1 and Even power 3.This is the pattern.
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Two approaches:
1:: Divisibility by 11
A number is divisible by 11 if the difference between the alternating sums of its digits = a multiple of 11. (For instance, 1243 divides by 11, because (2 + 3) - (4 + 1) = 0. 9273 is also divisible by 11, because (9 + 7) - (2 + 3) = 11, etc.)
10�� + 2 is a 50 digit number that starts with a 1, ends with a 2, is otherwise all 0s. So its alternating sum difference is 2 - 1 = 1. This means it has a remainder of 11, and we're done.
2:: Modular arithmetic
10 has a remainder of -1 when divided by 11, so 10�� has a remainder of (-1)��, or -1, when divided by 11. Since 10�� has a remainder of -1, 10�� + 2 has a remainder of -1 + 2, or 1.
1:: Divisibility by 11
A number is divisible by 11 if the difference between the alternating sums of its digits = a multiple of 11. (For instance, 1243 divides by 11, because (2 + 3) - (4 + 1) = 0. 9273 is also divisible by 11, because (9 + 7) - (2 + 3) = 11, etc.)
10�� + 2 is a 50 digit number that starts with a 1, ends with a 2, is otherwise all 0s. So its alternating sum difference is 2 - 1 = 1. This means it has a remainder of 11, and we're done.
2:: Modular arithmetic
10 has a remainder of -1 when divided by 11, so 10�� has a remainder of (-1)��, or -1, when divided by 11. Since 10�� has a remainder of -1, 10�� + 2 has a remainder of -1 + 2, or 1.