11 is a good divisor!

Number Theory Level 3

A number has 2014 digits in decimal representation, and leaves a remainder 5 when divided by 11.

What is the remainder when the same number formed by reversing it's digits is divided by 11?

Details and Assumptions

$\bullet$ Number formed by reversing digits of $12345$ is $54321$

$\bullet$ $0123$ is a 3-digit number, not $4$ digit.

This is a part of the set 11≡ awesome (mod remainders)

10 0 6 5

2 solutions

Trevor Arashiro
Sep 15, 2014

The divisibility by 11 rule is to add every other number starting from the first digit, then subtract every other number starting from the second and if the difference between the two is divisible by 11, the original number is divisible by 11.

Let us call the sum of every other number from the first a and from the second b. $a-b\equiv 5 \pmod{11} \therefore b-a\equiv -5 \equiv 6 \pmod{11}$ .

NOTE: if this number had an odd number of digits, then if the number was flipped the remainder wouldn't change. Eg abc: a+c-b, cba: c+a-b.

Good solution, just an addition that what you used for (mod 11) was type the same in LaTeX wrapping, but there is a code for that too, I've edited that in your solution. Please see the change made , see that from the "pencil" icon and then see how it works.

\pmod{11} appears as $\pmod{11}$

Aditya Raut - 6 years, 8 months ago

Ahh, yes. Thank you.

Trevor Arashiro - 6 years, 8 months ago

i got it.......its 5+6=11

Laika Mae Claus - 6 years, 8 months ago

Afreen Sheikh
Jan 22, 2015

One of the property of 11

when the digits are odd the remainder is going to be same when the no is reversed while

when the digits are even then the summation of remainder of original no and the reversed no gives 11

hence 11-5=6.

11 is a good divisor!

This is a part of the set 11≡ awesome (mod remainders)

2 solutions

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