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Counting Coins by Remainders (illustration)
FILE No.81Unsolved

Counting Coins by Remainders

Difficulty ★★☆about 6 minLogic PuzzleRemaindersNarrowing down
Fewer than 100 coins. Remainders: 2 when grouped by 3, 3 by 5, 2 by 7.

Case Record

In a thief's hideout, police found a bag of gold coins. The count is unknown, but the thief's notebook has these notes:

  • Dividing into groups of 3 leaves 2
  • Dividing into groups of 5 leaves 3
  • Dividing into groups of 7 leaves 2

The bag holds fewer than 100 coins.

Your question

How many coins were in the bag?

Case Files

Evidence

Diagram

Leaves 2 (by 3), 3 (by 5)CandidateRemainder by 781232 (match)383534685836980Only 23 leaves 2 when divided by 7.
Diagram

Timeline

  1. A bag of coins is found in a hideout.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

List the numbers that satisfy the first two conditions.

Open hint 2

Which of them leaves 2 when divided by 7?

Open hint 3

Remember the limit of 100.

Solve It

Your question

How many coins were in the bag?

Choose the answer closest to the truth.

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The Truth

23

The answer is: 23.

Numbers that leave 2 when divided by 3 and 3 when divided by 5 are 8, 23, 38, 53, 68, 83, 98 (they come every 15). Only 23 also leaves 2 when divided by 7 (23 = 7 x 3 + 2). The others leave 1, 3, 4, 5, 6 and 0.

Stack the conditionsBy 3 leaves 2 and by 5 leaves 3: every 15 (8, 23, 38, ...)Under 100: 8, 23, 38, 53, 68, 83, 98Only 23 leaves 2 by 7
Diagram of the solution

Key Points

  • Use two conditions first to build a short list
  • Use the third condition to pick one
  • Do not forget the range limit

Why people get fooled

Trying to satisfy all three conditions at once loses track of the clues.

Similar tricks

  • The write-out-and-narrow type
  • The one-condition-at-a-time type

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