The remainder is always less than the divisor

4.NBT.B.6 · take · grade 4

Archetype: Divisibility and Remainder Reasoning · step in a 8-type progression

The box $\square$ can be filled with any digit from $0$ to $9$ . In the division below, find the digit for $\square$ that makes the remainder as large as possible.

$78\square \div 27$

Show solution

Understand

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.

Givens

The dividend is 78□, where □ is one digit from 0 to 9
The divisor is 27
We want the remainder to be as large as possible

Unknowns

The digit □ that maximizes the remainder

Constraints

A remainder must be less than the divisor, so the remainder can be at most 26
The dividend ranges only from 780 to 789

Plan

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem

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.

Execute

#9 Solve an Easier Related Problem 4.NBT.B.6

Because the remainder is always smaller than the divisor, the biggest remainder possible when dividing by 27 is 26. We hope to hit exactly remainder 26.

\text{remainder} < 27 \Rightarrow \text{max remainder} = 26

Knowing the remainder caps at one less than the divisor turns this into a target hunt.

#6 Guess and Check 4.NBT.B.6

Find a multiple of 27 near 780. 27 x 28 = 756 and 27 x 29 = 783. To get remainder 26 we need a multiple of 27 plus 26 to land between 780 and 789: 756 + 26 = 782, which is in range.

27 \times 28 = 756,\quad 756 + 26 = 782

Adding the biggest remainder to a known multiple shows exactly which dividend works.

#6 Guess and Check 4.NBT.B.6

Since 782 = 78□ with □ = 2, choose □ = 2. Check: 782 ÷ 27 = 28 remainder 26, the largest possible remainder.

782 \div 27 = 28 \text{ R } 26

The dividend 782 sits in the allowed 780-789 window, so the digit is 2.

Answer: □ = 2 (then 782 ÷ 27 = 28 remainder 26)

Review

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.

Guess and Check (tool 6) every digit: divide 780 through 789 by 27 and list the remainders (24, 25, 26, 0, 1, ...); the largest, 26, occurs at 782, so □ = 2.

Standards · min grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing by 27 to compute quotients and remainders and using the remainder-less-than-divisor rule.

💡 This only needs the Grade 4 fact that a remainder is always smaller than the divisor -- aim for 26 and find the matching digit!