Couples in a Candy Store

  1. Three couples bought candy at a candy store. Each person paid as many cents per candy as pieces of candy he or she bought, and everyone bought a different number of pieces (no one was empty handed.) When they compared their bills, they found that the difference between the amounts paid by each husband and wife was the same. What is the smallest number of candies purchased? Show the amounts for the difference and for each person/couple.
  2. What if there were two couples?
  3. Four couples?
  4. Five couples?
  5. Can a method be extended for M couples?
  6. If the difference between each husband's and wife's amount is less than 500 cents, what was the largest number of couples possible in the store?

Source: 1. Somewhere I can't remember (if you find it, let me know...), 2-6: Original.

Solutions were received from MANY people.
  1. For 3 couples: (1,7) (4,8) (11,13). Difference is $0.48, total candies = 44.
  2. For 2 couples: (1,4) (7,8). Difference is $0.15. Total candies is 20.
  3. For 4 couples: (5,13) (9,15) (16,20) (35,37). Difference is $1.44. Total candies is 150.
  4. For 5 couples: (1,17) (6,18) (14,22) (21,27) (34,38). Difference it $2.88. Total candies is 198.
  5. In general for M couples, find a difference D with M pairs of factors: (B>A, A*B=D, A and B both odd or both even.) Then for each pair of factors A and B, the couple's values can be found as (B+A)/2 and (B-A)/2.
  6. For a difference less than 500 [cents], you can get 7 couples: (2,22) (7,23) (14,26) (19,29) (37,43) (58,62) (119,121). Difference is $4.80.
Solutions can be found for some of the answers above with a smaller difference, but some numbers of candies are then duplicated.
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