Problem 2 · Compare two division sentences · 3 points
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A positive whole number gives a quotient and a remainder when divided by 7. When the same number is divided by 8, the quotient and remainder switch places. What is the number?
- 55
- 62
- 47
- 54
- 56
Show a hint
- Picture the number as complete groups of 7 plus leftovers, then as complete groups of 8 plus leftovers. How do those two pictures compare?
- Call the first quotient q and remainder r. The two ways to build the number are 7q + r and 8r + q; also, r must be less than 7.
Show the answer and solution
Answer: A — 55
Since both expressions give the same number, 7q + r = 8r + q, so 6q = 7r. The only possible positive remainder less than 7 that makes this work is r = 6, giving q = 7. The number is 7 × 7 + 6 = 55, and 55 divided by 8 gives quotient 6, remainder 7. The tempting 47 gives quotient 6, remainder 5 when divided by 7; division by 8 gives quotient 5 but remainder 7, not 6.