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2022 Math Kangaroo

Problem 10

Problem 10 · 2022 Math Kangaroo Medium
Number Theory last-digitmod-10

Gerhard writes down the sum of the squares of two numbers, but some ink has run out so we cannot read every digit (see diagram). What is the last digit of the first number?

Figure for Math Kangaroo 2022 Problem 10
Show answer
Answer: C — 5
Show hints
Hint 1 of 2
You only need the LAST digit of the answer, so only the units digits of the two numbers matter.
Still stuck? Show hint 2 →
Hint 2 of 2
Squares can only end in 0, 1, 4, 5, 6 or 9; use the readable last digit of the total and of the second number.
Show solution
Approach: track only the units digit
  1. Only units digits matter: the visible total ends in 9, and the second number ends in 2, so its square ends in 4.
  2. Then the first number's square must end in 9 - 4 = 5, and a square ends in 5 only when its base ends in 5.
  3. So the last digit of the first number is 5, the answer is C.
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