Problem
The remainder is always less than the divisor
In the division 78□ ÷ 27, the box is a single digit 0-9 making 78□ a three-digit number between 780 and 789. Choose the digit that makes the remainder as large as it can be.
Base-ten numbers
Your answer
Watch out Don't assume the remainder can reach the divisor itself (27) — it's always less than the divisor, so the true maximum here is 26.
How to solve
Strategy Guess and Check — The remainder when dividing by 27 can be at most 26, so we look for a dividend in the range 780-789 that leaves remainder 26. Finding the nearest multiple of 27 below this range and adding 26 turns it into a simple check of a single digit.
1STEP 1
Find the largest possible remainder
A remainder is always smaller than the divisor, so dividing by 27 gives at most 26 — aim to hit that exactly.
remainder less than 27 → max remainder = 26
Knowing the remainder caps at one less than the divisor turns this into a target hunt.
4.NBT.B.6Solve An Easier Related ProblemThe largest remainder possible when dividing 78□ by 27 is 26.
Why?
Dividing 78□ by 27 breaks it into a whole number of complete groups of 27 plus one leftover, and that leftover is the remainder.
Why?
The complete groups and the leftover, put back together, are exactly 78□ with nothing missing and nothing counted twice.
🧱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?
That leftover can never be 27 or more, so the biggest it can reach is one less, 26.
Why?
A complete group of 27 needs a full 27 items, so if 27 or more were still left you could form one more whole group and the leftover would drop below 27.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Locate the multiple of 27 just below the range
27 × 28 = 756, so 756 + 26 = 782 — right in the 780-789 range.
27 × 28 = 756, 756 + 26 = 782
Adding the biggest remainder to a known multiple shows exactly which dividend works.
4.NBT.B.6Guess And Check3STEP 3
Read off the digit
Since 782 = 78□ with □ = 2, choose □ = 2. Check: 782 ÷ 27 = 28 remainder 26, the largest possible remainder.
782 ÷ 27 = 28 R 26
The dividend 782 sits in the allowed 780-789 window, so the digit is 2.
4.NBT.B.6Guess And CheckAnswer
2
756 + 26 = 782, so □ = 2
Remainder 26 is less than the divisor 27, which is required, and it is the maximum allowed. The dividend 782 is within 780-789, so the digit 2 is valid. The next dividend 783 gives remainder 0, confirming 782 is the right choice.
Takeaway
This only needs the Grade 4 fact that a remainder is always smaller than the divisor -- aim for 26 and find the matching digit!
- Find the largest possible remainder
- Locate the multiple of 27 just below the range
- Read off the digit
Where next?
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