Maths Olympiad Prep

Library / /95 of 173

Algebra Difficulty 2.7 Junior Find the answer

When kk candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, 3k3 k candies were distributed among seven people in this way, then how many candies would have been left over?

A number or a short expression. Spacing and $ signs are ignored.

Solution

Suppose that each of the 7 people received qq candies under the first distribution scheme. Then the people received a total of 7q7 q candies and 3 candies were left over. Since there were kk candies, then k=7q+3k=7 q+3. Multiplying both sides by 3, we obtain 3k=21q+93 k=21 q+9. When 21q+921 q+9 candies were distributed to 7 people, each person could have received 3q+13 q+1 candies, accounting for 21q+721 q+7 candies in total, with 2 candies left over.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.