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This essential trains for: SAT-I, GMAT, Math Kangaroo 7-8, Math Kangaroo 9-10.

The sum of the digits of a positive integer written in base 10 has a number of interesting properties.

Denote the sum of the digits of N with:

equation

For an integer with m digits, the sum of the digits cannot exceed:

equation

and cannot be less than:

equation

Take a and b, arbitrary integers. The sums of their digits are:

equation

What is the sum of the digits of a+b?

As long as there are no carry-overs, the sum of the digits of the sum is equal to the sum of the sums of digits:

equation

This is a consequence of the commutative property of addition (we can add the digits in any order).

However, if a carry-over happens, then the sum of the digits that exceeds 1 will be diminished by 10 (since the maximum carry for adding two digits is 1), while the sum of the digits of the next higher place value will be increased by 1. Therefore, the effect of a carry-over on the sum of the digits is that of subtracting a 9.

Example:

equation

The sum of the digits of a number has the same remainder as the number itself with respect to division by 3 and by 9.

This is because:

equation

equation

Therefore,

equation

equation

Expanding the number in base 10:

equation

equation

It is obvious that this property is valid only in base 10.