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Algebra Arithmetic Math Olympiad PRMO

Sum of Digits base 10 | PRMO II 2019 | Question 7

Try this beautiful problem from the PRMO II, 2019 based on the Sum of Digits base 10. You may use sequential hints to solve the problem.

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.

Source
Suggested Reading

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.

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