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Difficulty: MediumFactors, Multiples, and Prime Factorization

A youth soccer league has a group of players. The players can be divided into equal-sized teams of either 12 or 18, with no players left over. However, if they are divided into teams of 15, there are exactly 3 players left over. What is the least possible number of players in the league?

  1. A
    3
  2. B
    36
  3. C
    72
  4. 108Answer
  5. E
    180

Answer

The least possible number of players in the league is 108.
To satisfy the condition that players can be divided into teams of 12 or 18 with no remainders, the total number of players must be a multiple of their least common multiple (LCM). The prime factorization of 12 is 22×32^2 \times 3, and for 18 it is 2×322 \times 3^2, so their LCM is 22×32=362^2 \times 3^2 = 36. Next, we evaluate the positive multiples of 36 to find the smallest one that leaves a remainder of 3 when divided by 15. Testing 36: 36=15(2)+636 = 15(2) + 6 (remainder 6). Testing 72: 72=15(4)+1272 = 15(4) + 12 (remainder 12). Testing 108: 108=15(7)+3108 = 15(7) + 3 (remainder 3). Thus, 108 satisfies all conditions.

Step-by-Step Solution

1
Find the least common multiple (LCM) of 12 and 18.
The LCM of 12 and 18 is 36.
Since the players can be divided into teams of 12 or 18 without remainders, the total number of players must be a multiple of both 12 and 18, which means it must be a multiple of their LCM.
2
List the positive multiples of 36.
The multiples are 36, 72, 108, 144, 180, and so on.
To find the least possible number of players, we need to test the multiples of 36 in increasing order.
3
Test the multiples of 36 to find which one leaves a remainder of 3 when divided by 15.
Testing 36 gives a remainder of 6; testing 72 gives a remainder of 12; testing 108 gives a remainder of 3.
Dividing 108 by 15 yields 7 with a remainder of 3, satisfying all the conditions of the problem.

Key Concept

Using the least common multiple (LCM) to satisfy multiple divisibility and remainder constraints.
Estimated Time:1m 30s
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Topics

Factors, Multiples, and Prime Factorization

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