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Number properties

This topic has 2 expert replies and 5 member replies

Number properties

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What is the remainder when 10^ 49 +2 is divided by 11 ?

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sud21 wrote:
What is the remainder when 10^ 49 +2 is divided by 11 ?

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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.

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Brent

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It should be read as What is the remainder when {10^49} +2 is divided by 11 ?

Can it be solved?

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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
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.
So on.
SO 10^49+2 will have remainder of 1.Ans-A

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That's what I thought. You divide any 1000 or 10000, if unit digit ends in zero, then the maximum way the quotient can be is 9 and then you add 2 which becomes 3.

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If u have noticed :Here every Odd power of 10 would produce remainder of 1 and Even power 3.This is the pattern.

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The pattern is like that here.

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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.)

10⁴⁹ + 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 10⁴⁹ has a remainder of (-1)⁴⁹, or -1, when divided by 11. Since 10⁴⁹ has a remainder of -1, 10⁴⁹ + 2 has a remainder of -1 + 2, or 1.

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