Problem 20 · 2019 Math Kangaroo
Stretch
Number Theory
divisibility
Benjamin writes a number into the first circle. He then carries out the calculations shown along the arrows and each time writes the result in the next circle. How many of the six numbers are divisible by 3?
Show answer
Answer: B — 2
Show hints
Hint 1 of 3
The starting number is unknown, so try a few different starting numbers and watch which results land on multiples of 3.
Still stuck? Show hint 2 →
Hint 2 of 3
Among any three numbers in a row, exactly one is a multiple of 3.
Still stuck? Show hint 3 →
Hint 3 of 3
Check whether the count of multiples of 3 stays the same no matter where you start.
Show solution
Approach: try a couple of starting numbers and spot the steady pattern
- Pick an easy start and follow the arrows; then pick a different start and do it again, each time circling the results that are multiples of 3.
- Both times you find exactly two multiples of 3: one comes from the first three numbers (one of any three numbers in a row is always a multiple of 3), and one comes from the step that multiplies by 3.
- The last two results are always 2 more, and double of that, so they are never multiples of 3, leaving exactly 2 divisible by 3 (B).
Why it is always exactly 2 (algebra)
If the start is n, the six numbers are n, n+1, n+2, 3(n+2), 3(n+2)+2, and 6(n+2)+4. Exactly one of n, n+1, n+2 is divisible by 3, 3(n+2) always is, and the last two leave remainders 2 and 1, so the count is always 2.
Mark:
· log in to save