Invariance Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Show that the sum of the digits of a multiple of 9 is always a multiple of 9. Verify with

n = 198

and

n = 729

1
Any integer $n$ can be written as $n = \sum_i a_i \cdot 10^i$ where $a_i$ are digits. Since $10 \equiv 1 \pmod{9}$ , we get $n \equiv \sum_i a_i \pmod{9}$ .
2
So $9 \mid n \Leftrightarrow 9 \mid (\text{sum of digits})$ — the divisibility by 9 is an invariant property shared by $n$ and its digit sum.
3
Check $n=198$ : digit sum $= 1+9+8 = 18$ , which is a multiple of 9. And $198 = 9 \times 22$ . Confirmed.
4
Check $n=729$ : digit sum $= 7+2+9 = 18$ , multiple of 9. And $729 = 9 \times 81$ . Confirmed.

Answer

9 \mid n \Leftrightarrow 9 \mid (\text{digit sum of }n)

An invariant is a property preserved across transformations. Here, taking digit sums preserves divisibility by 9 because of how 10 behaves modulo 9.

About Invariance

A property of a mathematical object that remains unchanged when the object undergoes a particular transformation or operation.

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