MathLabs

Problem 1

For a positive integer n, let S(n) be the sum of the digits in the decimal representation of n. A positive integer obtained by removing at least one digit from the right-hand end of n is called a stump of n. Let T(n) be the sum of all stumps of n. Prove that n=S(n)+9T(n).
Step 2 of 4: Subtract the digit sum
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m−S(m)=10n−S(n)m-S(m)=10n-S(n)
Detailed analysis

Using m=10n+a and S(m)=S(n)+a, the appended digit cancels: m-S(m)=10n+a-S(n)-a=10n-S(n).