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Reverse Equivalence

Problem

By adding the different 2-digit numbers 12 and 32 we get 44. If the digits in each number are reversed we get two different 2-digit numbers, and 21 + 23 also equals 44.
The same is true of 42 + 35 = 24 + 53 = 77.

Prove that the sum of two 2-digit numbers with this property will always be divisible by 11.


Solution

Let the 2-digit numbers be (ab) and (cd).

We shall ignore cases like 11 + 11 = 11 + 11 and 12 + 21 = 21 + 12, as proving that (ab) + (ba) = 10a + b + 10b + a = 11a + 11b is divisibly by 11 is trivial.

The problem requires the sum, S = (ab) + (cd) = (ba) + (dc), such that (ab) not equal (cd) not equal (ba)

S = (10a + b) + (10c + d) = (10b + a) + (10d + c).

Therefore 9a + 9c = 9b + 9d and so a + c = b + d.

Hence we obtain a pair of numbers with the required property if the sum of the first digits of each number is equal to the sum of the second digits. For example, 41 + 36 = 14 + 63 = 77, because 4 + 3 = 1 + 6.

Now S = (10a + b) + (10c + d) = 10(a + c) + b + d

As a + c = b + d, S = 10(a + c) + a + c = 11(a + c).

Hence the sum, S, is always divisible by 11.

What adding three 2-digit numbers?
Can you find two 3-digit numbers with this property?

Problem ID: 126 (Oct 2003)     Difficulty: 3 Star

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