Try this beautiful problem from the PRMO II, 2019 based on Sum of Digits base 10.

## Sum of Digits base 10 – PRMO II 2019

Let s(n) denote the sum of the digits of a positive integer n in base 10. If s(m)=20 and s(33m)=120, what is the value of s(3m)?

- is 107
- is 60
- is 840
- cannot be determined from the given information

**Key Concepts**

Real Numbers

Algebra

Integers

## Check the Answer

But try the problem first…

Answer: is 60.

PRMO II, 2019, Question 7

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

taking sum of digit base 10 to (mod 9)

and s(ab)=s(a).s(b)(mod 9)

[ let x congruent r mod n, y congruent to s mod n,

\(0 \leq r,s \leq n-1\),

x=in+r, y=jn+s, i,j are integers

xy=(in+r)(jn+s)=ij\(n^2\)+(is+jr)n+rs congruent to rs mod n

so, xy mod n =(x mod n)(y mod n) ]

Second Hint

given s(m)=20

s(33m)=120=\(s(11) \times s(3m)\)

or, 120=\(2 \times s(3m)\) [ since s(11)=2(mod 9)]

Final Step

or, 60=s(3m)

so, s(3m)=60.

## Other useful links

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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