Recover missing digits from column carries

3.NBT.A.33.OA.B.5 · take · grade 3

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

In the multiplication below, find the digits $A$ and $B$ .

\begin{array}{r} A\ B \\ \times\quad 6 \\ \hline 2\ 8\ 8 \end{array}

Here $A$ is the tens digit and $B$ is the ones digit of the two-digit number being multiplied. The two-digit number times $6$ equals $288$ .

Show solution

Understand

A two-digit number with tens digit A and ones digit B is multiplied by 6, and the product is 288. We must find the digits A and B.

Givens

The two-digit number is written A B (A tens, B ones)
The two-digit number times 6 equals 288

Unknowns

The tens digit A
The ones digit B

Constraints

A and B are single digits (0-9), and A is not 0 since A B is a two-digit number

Plan

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

The result of the multiplication is given, so we undo it: dividing 288 by 6 recovers the two-digit number, and reading its digits gives A and B. A quick column check confirms the carry.

Execute

#11 Work Backwards 3.OA.A.3

Since the two-digit number times 6 equals 288, divide 288 by 6 to recover the number.

288 \div 6 = 48

Multiplication and division are opposites, so dividing the product by 6 gives back the original number.

#11 Work Backwards 3.NBT.A.3

The recovered number is 48, so the tens digit A is 4 and the ones digit B is 8.

48 \Rightarrow A = 4,\ B = 8

Place value tells us the left digit is the tens and the right digit is the ones.

#6 Guess and Check 3.OA.B.5

Verify by multiplying: 6 times 8 = 48, write 8 carry 4; 6 times 4 = 24, plus the carry 4 = 28. That builds 288.

48 \times 6 = 288

Rebuilding the product column by column confirms the carries line up exactly.

Answer: A = 4, B = 8

Review

48 times 6 = 288 exactly, and both A=4 and B=8 are valid single digits with A not zero, so the two-digit number 48 is legitimate.

Guess and check the ones column first (tool 6): 6 times B must end in 8, so B = 3 or 8; only B = 8 with the carry produces 288, then the tens column forces A = 4.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 288 by 6 to recover the two-digit factor
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Using place value to read the tens digit A and ones digit B
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Checking the column multiplication with carrying

💡 When you know the answer to a times problem, just divide to walk backwards to the hidden number, then read its digits!