Depth · one concept from first principles · step 2 of 4
Divisor of dividend minus remainder, within bounds
Dividing 51 by Mia's age leaves a remainder of 13, and Mia is younger than her 22-year-old sister. Find Mia's age.
OperationsExpressions
Your answer
How to solve
Strategy Work Backwards — Undo the division: taking the remainder away leaves an exact multiple of Mia's age. List the divisors of that number, then keep only the one that fits between the two bounds.
1STEP 1
Take the remainder away
If Mia is x years old and the quotient is Q, then 51 = x × Q + 13. Removing the 13 left over leaves a number that x divides exactly.
51 - 13 = 38 = x × Q
Mia's age must be a divisor of 38.
4.OA.B.4Work BackwardsMia's age divides exactly.
Why?
Dividing by Mia's age means is some full groups of her age plus a leftover of .
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
Why?
Taking the same away from both sides keeps them equal, so the full groups alone make .
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
2STEP 2
List the divisors of 38
Write 38 as a product in every way.
38 = 1 × 38 = 2 × 19
The divisors are 1, 2, 19, 38.
4.OA.B.4Make A Systematic List3STEP 3
Apply both bounds
The divisor must be bigger than the remainder 13, and Mia is younger than 22. Only one divisor sits in that range.
13 ＜ x ＜ 22 → x = 19
Mia is 19.
7.EE.B.4Eliminate PossibilitiesAnswer
19 years old
Check: 19 × 2 + 13 = 51, and 13 ＜ 19, so 51 divided by 19 really leaves 13. Also 19 ＜ 22.
Takeaway
Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.
- Take the remainder away
- List the divisors of 38
- Apply both bounds
Where next?
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