Maths Fun with Digit Sums

MATHS FUN WITH DIGIT SUMS

The digit sum of a given number is the sum of all the digits of that number

e.g.,

The DIGIT SUM of12345 = 1 + 2 + 3 + 4 + 5 = 15

Q. Find the biggest number, less than 100, whose DIGIT SUM is 11?

First let’s look at the pairs of numbers that add up to 11.

1 + 102 + 93 + 84 + 75 + 6

10 + 1 9 + 2 8 + 3 7 + 4 6 + 5

Q. Find the biggest number, less than 100, whose DIGIT SUM is 11?

Let us now use each pair to get a possible answer.

1 + 102 + 93 + 84 + 75 + 6

10 + 1 9 + 2 8 + 3 7 + 4 6 + 5

11029384756

101 92 83 74 65

We can see that 110 and 101 won’t work because their digit sums are 2 (1 + 1 +

0).

Our answer is therefore 92.

1 + 102 + 93 + 84 + 75 + 6

10 + 1 9 + 2 8 + 3 7 + 4 6 + 5

11029384756

101 92 83 74 65

But my mind would not stop there!!I began adding up the numbers and

discovered a pattern:

29 + 92 = 12138 + 83 = 12147 + 74 = 12156 + 65 = 121

The pattern explained

with an equation is this:

10a + b + 10b + a = 11a + 11b = 11(a + b)

So for any number ab

ab + ba = 11(a + b)

And the mind still did not stop!!!So I started looking at 3 digit numbers

and saw the following pattern.

For any number abcabc + bca + cab = 111(a + b + c)

The pattern explained:100a + 10b + c + 100b + 10c + a + 100c +

10a + b = 111a + 111b + 111c

= 111(a + b +c)

An example:562 + 625 + 256 = 1443

The Digit Sum of 562 is 5 + 6 + 2 = 13111 x 13 = 1443

Proving that 562 + 625 + 256 = 111(5 + 6 + 2)

Let us test the pattern with a 3 digit number.

Have fun testing the patterns shown with other 2 digit and 3 digit numbers and

perhaps discover other patterns!

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