Problem 14 · 2012 Math Kangaroo
Medium
Number Theory
mod-10divisibility
The natural numbers are to be painted: 1 is red, 2 is blue, 3 is green, 4 is red, 5 is blue, 6 is green, and so on. Which colour(s) can the sum of a red number and a blue number have?
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Answer: A — green only
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Hint 1 of 2
The colours repeat every three numbers: red, blue, green, red, blue, green...
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Hint 2 of 2
Think about the remainders when red and blue numbers are divided by 3, then add them.
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Approach: use remainders mod 3
- Red numbers are 1, 4, 7, ... (remainder 1 when divided by 3); blue are 2, 5, 8, ... (remainder 2).
- A red plus a blue number has remainder 1 + 2 = 3, that is remainder 0 (a multiple of 3).
- Multiples of 3 (3, 6, 9, ...) are exactly the green numbers, so the sum is always green only.
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