Problem
Numbers whose quotient equals the remainder
A three-digit number is divided by 73. The quotient and the remainder must come out equal. A remainder is always less than 73. Count how many three-digit numbers do this.
Reasoning · Grade 4-1 Division
Your answer
Watch out It looks like multiples of 73, but combining the two gives multiples of 74.
How to solve
Strategy Make a Systematic List — Testing all 900 three-digit numbers one by one is far too slow, so I flip the division round: instead of starting from the number and dividing, I start from the shared value of the quotient and remainder and rebuild the number. That value is a genuinely finite list — it can only be 0 up to 72, because a remainder never reaches the divisor — and each choice builds exactly one number. Writing the first few of those numbers in order shows they climb by a fixed step, so I can list the candidates in order and cross off the ones that are not three-digit numbers.
1STEP 1
Write the division as a multiplication
Write it as 73 × quotient + remainder.
number = 73 × quotient + remainder
This is the same check students already do after a long-division exercise, just read forwards instead of backwards.
4.NBT.B.6Make A Systematic List2STEP 2
Use the condition to collapse the two unknowns into one
Since they are equal the number is a multiple of 74.
73 × □ + □ = 74 × □
Seventy-three of something plus one more of the same something is seventy-four of it — the same 'collect like groups' idea used with 73 apples plus 1 apple.
4.OA.B.4Look For A PatternBecause the quotient and the remainder are the same number, seventy-three of it plus one more of it is seventy-four of it.
Why?
Groups of the same thing can be collected together, so a shared factor may be pulled out of the whole sum at once.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
The dividend is the divisor times the quotient plus the remainder, which is what lets the two unknowns be written as one.
🧱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.
3STEP 3
Bound the seed with the remainder rule
The remainder rule narrows the range.
If the remainder ever reached 73 you could take another whole group out, so it has to stay below 73 — that single rule turns an infinite hunt into a short list.
4.NBT.B.6Eliminate Possibilities4STEP 4
List the candidates and see the pattern
The candidates run 74, 148, 222, and so on.
0, 74, 148, 222, 296, 370, …
Because the list climbs by the same 74 every time, you only have to find where it enters the three-digit range and where it leaves it.
4.NBT.B.5Make A Systematic List5STEP 5
Cross off the seeds that fail the three-digit test
The three-digit ones run 148 to 962.
74 × 2 = 148, 74 × 13 = 962, 74 × 14 = 1036
Checking only the two edges of the range is enough, because everything between them is automatically between 148 and 962.
4.NBT.B.5Eliminate Possibilities6STEP 6
Count the surviving seeds
Counting them gives 12.
13 - 2 + 1 = 12
Counting whole numbers from 2 to 13 needs the 'add one back' rule, because subtracting alone would forget the very first one in the list.
4.OA.A.3Make A Systematic List7STEP 7
Test one answer to be sure
Dividing 592 gives quotient and remainder both 8.
592 ÷ 73 = 8 R 8, 73 × 8 + 8 = 592
Spot-checking one number from the middle of the list confirms the multiplication shortcut really does reproduce the original division.
4.NBT.B.6Make A Systematic ListAnswer
12 numbers
13 − 2 + 1 = 12
The answer counts numbers, so it must be a whole number no bigger than the 900 three-digit numbers there are, and 12 is comfortably in range. It also makes sense as a rough size: the qualifying numbers are exactly the multiples of 74 with three digits, and 900 divided by 74 is about 12, which matches. Every listed number is a genuine three-digit number, and the largest, 962, is under 999 while the next one up, 1036, is over — so the list is neither short nor long by one.
Takeaway
When the quotient and the remainder are the same, 73 groups plus 1 leftover is really 74 groups — so the answers are just the three-digit multiples of 74.
- Write the division as a multiplication
- Use the condition to collapse the two unknowns into one
- Bound the seed with the remainder rule
- List the candidates and see the pattern
- Cross off the seeds that fail the three-digit test
- Count the surviving seeds
- Test one answer to be sure