Problem
Smallest number meeting remainder bounds
A whole number lies strictly between 80 and 90. Divided by 5 it leaves 2. Divided by 7 it leaves 3. Find it.
Reasoning · Grade 3-2 Conditions and Division
Your answer
Watch out Only ones digits of 2 come to mind, but leaving 2 after dividing by 5 also allows a ones digit of 7.
How to solve
Strategy Make a Systematic List — The range 81-89 is tiny. The 'remainder 2 when divided by 5' clue means the number ends in 2 or 7, which leaves only two candidates; I then test those two against the divide-by-7 clue.
1STEP 1
Use the divide-by-5 remainder to narrow the list
Leaving 2 after 5 needs a ones digit of 2 or 7: 82 and 87.
82 = 5 × 16 + 2, 87 = 5 × 17 + 2
Remainder 2 after dividing by 5 just means 'two past a multiple of 5,' so the ones digit must be 2 or 7 - that is a fast filter.
3.OA.C.7Make A Systematic ListLeaving remainder 2 after dividing by 5 means the number sits two past a multiple of 5, so its last digit is 2 or 7.
Why?
Splitting the number into whole fives plus a leftover is always possible in exactly one way, and here the leftover is fixed at 2.
🧱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?
Since every ten is a multiple of 5, only the ones digit can decide the leftover, so the leftover pins the last digit.
🧱Divisibility tests come from place valueBecause 10, 100 and 1000 sit in a fixed relation to a divisor, only certain digits matter — the last two decide 4, the digit sum decides 3 and 9.
2STEP 2
Test each candidate against the divide-by-7 clue
By 7, 82 leaves 5 while 87 leaves 3.
82 = 7 × 11 + 5, 87 = 7 × 12 + 3
Dividing with a remainder is a Grade 3 skill; comparing the leftover to 3 immediately tells me which candidate fits.
3.OA.B.6Eliminate Possibilities3STEP 3
Confirm the range
It also sits in range, so the answer is 87.
80 ＜ 87 ＜ 90
Checking the survivor against every condition confirms it is the one and only number that works.
3.OA.A.4Eliminate PossibilitiesAnswer
87
87 = 5 × 17 + 2 = 7 × 12 + 3
87 divided by 5 gives 17 remainder 2, and 87 divided by 7 gives 12 remainder 3, and 87 lies between 80 and 90. The other candidate (82) failed the divide-by-7 test, so 87 is the unique answer.
Takeaway
Remainder 2 after dividing by 5 means the ones digit is 2 or 7 - that one trick shrinks the whole search to just two numbers to test!
- Use the divide-by-5 remainder to narrow the list
- Test each candidate against the divide-by-7 clue
- Confirm the range