Problem
Subtract the remainder to find a divisor
I need the biggest number that leaves a remainder of 4 when it divides 148, and also leaves a remainder of 5 when it divides 165.
OperationsBase-ten numbers
Your answer
How to solve
Strategy Identify Subproblems — The key idea is that if a remainder is left over, removing that remainder makes the number divide evenly. So I break the task into subproblems: turn each 'remainder' clue into an 'exact divisor' clue by subtracting the remainder, then look for the largest number that divides both results. I confirm the winner by guess-and-check against the original two conditions.
1STEP 1
Turn the first clue into an exact-division clue
Taking the remainder off 148 leaves 144, which the mystery number divides exactly.
148 - 4 = 144
A remainder is the leftover that won't fit; subtract it and what's left fills the groups perfectly, so the number is a factor of 144.
4.NBT.B.6Identify SubproblemsThe mystery number divides 144 exactly.
Why?
148 breaks into whole groups of the mystery number plus a leftover of 4, so taking that 4 away leaves nothing but the whole groups.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
What is left is made only of complete equal groups, which is exactly what dividing with no remainder means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Turn the second clue into an exact-division clue
The same move on 165 leaves 160, so the number is a factor of that one too.
165 - 5 = 160
The same 'remove the leftover' trick turns the second messy clue into a clean factor clue: the number is a factor of 160.
4.NBT.B.6Identify Subproblems3STEP 3
Find the common factors of 144 and 160
144 = 16 x 9 and 160 = 16 x 10, so the greatest number dividing both is 16.
144 = 16 × 9, 160 = 16 × 10
I want a factor shared by both numbers, and the biggest one; 16 splits 144 into 9 groups and 160 into 10 groups, with nothing left over.
4.OA.B.4Guess And Check4STEP 4
Check the size rule and confirm with the originals
A divisor has to exceed its remainders, and 16 does: 148 and 165 leave 4 and 5 as required.
148 ÷ 16 = 9 R 4, 165 ÷ 16 = 10 R 5
A remainder only makes sense if the divisor is bigger than it; 16 passes that test and reproduces both given remainders exactly.
4.NBT.B.6Guess And CheckAnswer
16
16 is larger than both remainders (4 and 5), which is required for the remainders to be valid. Re-dividing confirms 148 = 16 x 9 + 4 and 165 = 16 x 10 + 5, so both conditions are met and 16 is the largest such common factor.
Takeaway
Take away the leftover, and a tricky 'remainder' clue turns into a neat 'factor' clue.
- Turn the first clue into an exact-division clue
- Turn the second clue into an exact-division clue
- Find the common factors of 144 and 160
- Check the size rule and confirm with the originals
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested