Remainder must be less than the divisor

3.OA.B.63.OA.C.7 · take · grade 3

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

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 8 = 11 \cdots \blacktriangle$

Show solution

Understand

A number (the blank) divided by 8 gives quotient 11 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 8 has quotient 11 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 8.
The quotient stays exactly 11.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Use dividend = 8 times quotient plus remainder. To make the dividend largest while keeping quotient 11, choose the biggest allowed remainder, which is 7.

Execute

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 8, so the greatest it can be is 7. A remainder of 8 or more would increase the quotient.

\blacktriangle < 8 \Rightarrow \blacktriangle = 7

The remainder is always less than the divisor, so 7 is the biggest value that keeps the quotient at 11.

#11 Work Backwards 3.OA.C.7

The dividend equals 8 times the quotient 11, plus the remainder 7.

8 \times 11 + 7 = 88 + 7 = 95

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 11.

Answer: 95

Review

Check: 95 divided by 8 is 11 remainder 7, since 8 times 11 is 88 and 95 minus 88 is 7, and 7 is less than 8. Trying 96 would give quotient 12, too big, so 95 is the greatest.

List the candidates (tool 2): dividends giving quotient 11 are 88 (r 0) up to 95 (r 7); the next, 96, bumps the quotient to 12, so 95 is the maximum.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 8, so its largest value is 7.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 8 times 11 plus 7 to get the greatest dividend.

💡 This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!