Invariants Math Example 3

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Example 3

medium

The sum of digits of a number doesn't change modulo 9 when you add 9. Verify: 47 → 47+9=56. Is the digit sum invariant mod 9?

Solution

1
Compute the digit sum of 47: $4 + 7 = 11$ . Then $11 \mod 9 = 2$ .
2
Compute the digit sum of 56: $5 + 6 = 11$ . Then $11 \mod 9 = 2$ .
3
Both digit sums are congruent to 2 mod 9. This works because adding 9 to any number doesn't change its residue mod 9 (since $9 \equiv 0 \pmod{9}$ ), and a number's digit sum is always congruent to the number itself mod 9.

Answer

Yes, the digit sum mod 9 is invariant under adding 9. Both 47 and 56 have digit sum

\equiv 2 \pmod{9}

The digit sum modulo 9 is a classic invariant known as 'casting out nines.' Any number

n \equiv

(sum of its digits)

\pmod{9}

, and since

9 \equiv 0 \pmod{9}

, adding 9 preserves this residue. This invariant is useful for checking arithmetic.

About Invariants

Quantities or properties that remain unchanged during a process, operation, or transformation—values that stay the same no matter how the system is rearranged or acted upon.

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Example 2 hard

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Example 4 medium

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Example 5 hard

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