NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Spatial problem-solving algorithms

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A sports equipment supplier in Perth is allocating 61 identical rugby balls and 85 identical cricket balls to a number of regional training hubs. Each hub must receive the same number of rugby balls and the same number of cricket balls, with no balls left over.

The supplier uses a digital algorithm to determine the maximum number of balls of each type that can be evenly distributed.

What is the exact number of rugby balls each hub receives?

Concept illustrations

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Worked solution

  1. The supplier has 61 rugby balls and 85 cricket balls to distribute evenly among regional hubs with no balls left over.
  2. To find the maximum number of hubs that can receive the same number of each ball type, we need the greatest common divisor (GCD) of 61 and 85.
  3. 61 is a prime number because it has no divisors other than 1 and itself. We check if 61 divides 85.
  4. 85 ÷ 61 = 1.393... which is not an integer, so 61 does not divide 85.
  5. Since 61 is prime and does not divide 85, the only common divisor of 61 and 85 is 1.
  6. Therefore, the GCD of 61 and 85 is 1.
  7. This means the maximum number of hubs that can receive an equal number of each ball type is 1.
  8. With 1 hub, all 61 rugby balls are allocated to that single hub.
  9. The number of rugby balls each hub receives is 61 ÷ 1 = 61.
  10. Check: 61 ÷ 1 = 61 (integer), 85 ÷ 1 = 85 (integer), so no balls are left over.
  11. The solution satisfies the condition that each hub receives the same number of each ball type with no remainder.
  12. Therefore, the exact number of rugby balls each hub receives is 61.

Reference answer: 61

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