Fill long-division blanks from the known

3.OA.C.73.OA.B.6 · take · grade 3

Archetype: Recover Hidden Digits from Carries · step in a 5-type progression

Fill in each $\square$ with the correct digit.

The figure below is a long division. The divisor is 4, and the dividend is a two-digit number whose tens digit is 6 and whose ones digit is hidden by a $\square$ . The quotient is a two-digit number whose tens digit is hidden by a $\square$ and whose ones digit is 7. In the worked layout, the partial product of the tens digit (a one-digit number), the two-digit number brought down after subtracting, and the final two-digit number being subtracted each have their digits hidden by $\square$ , and the final remainder is 1. Find every hidden digit.

Show solution

Understand

In a long-division layout, 4 divides a two-digit dividend 6-blank, giving a two-digit quotient blank-7 with remainder 1. We must find every hidden digit, including the partial-product and subtraction rows.

Givens

The divisor is 4.
The dividend is 6 in the tens place and a hidden digit in the ones place.
The quotient is a hidden tens digit and 7 in the ones place.
The final remainder is 1.
The figure shows a one-box partial product, a two-box bring-down row, and a two-box subtraction row.

Unknowns

The dividend's ones digit, the quotient's tens digit, and every hidden box in the worked layout.

Constraints

Dividend = divisor times quotient plus remainder.
Every box is a single digit; the remainder 1 is less than the divisor 4.

Plan

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

Use the rule dividend = 4 times quotient plus 1 to recover the dividend, then replay the long division step by step to fill each hidden box.

Execute

#11 Work Backwards 3.OA.B.6

The dividend equals 4 times the quotient plus the remainder 1. The quotient ends in 7 and the dividend is in the 60s, so the quotient is 17, giving dividend 4 times 17 plus 1.

4 \times 17 + 1 = 68 + 1 = 69

Multiplying back from quotient and remainder is the inverse of dividing, so it pins down the dividend exactly.

#6 Guess and Check 3.OA.C.7

The dividend is 69, so its hidden ones digit is 9. The quotient is 17, so its hidden tens digit is 1.

69 \div 4 = 17 \cdots 1

Once the dividend is 69, reading its ones digit and the quotient's tens digit is direct.

#11 Work Backwards 3.OA.C.7

First 4 goes into 6 once. The quotient tens digit 1 times divisor 4 is 4, the one-box partial product. Subtracting 4 from 6 leaves 2.

1 \times 4 = 4,\quad 6 - 4 = 2

Each long-division step multiplies the new quotient digit by the divisor, so the partial product must be 4.

#6 Guess and Check 3.OA.C.7

Bring down the 9 to make 29, the two-box bring-down row. The ones quotient digit 7 times 4 is 28, the two-box subtraction row. Subtracting gives the remainder 1.

29 - 28 = 1,\quad 7 \times 4 = 28

The bring-down row 29 and the product 28 differ by exactly the remainder 1, confirming all boxes.

Answer: Dividend 69 and quotient 17: dividend ones digit = 9, quotient tens digit = 1, tens partial product = 4, bring-down row = 29, final subtraction row = 28 (remainder 1).

Review

Check by dividing: 69 divided by 4 is 17 remainder 1, since 4 times 17 is 68 and 69 minus 68 is 1, and 1 is less than 4. All digits are consistent.

Convert to a missing-factor equation (tool 13): solve 4 times Q plus 1 = 6_ where Q ends in 7; only Q = 17 keeps the dividend two digits starting with 6.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recovering the dividend 69 from the quotient and remainder by multiplying back.
3.OA.C.7 Fluently multiply and divide within 100 — Replaying each long-division step to fill the partial-product and subtraction boxes.

💡 This only needs Grade 3 division facts: multiply the quotient back to get the dividend, then redo the steps to fill every box!