Deduce missing digits in a multiplication

3.OA.C.73.NBT.A.3 · take · grade 3

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

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

$A74 \times B = 6992$

Here $A74$ is a three-digit number whose hundreds digit is $A$ , tens digit is $7$ , and ones digit is $4$ , and $B$ is a single digit.

Show solution

Understand

A three-digit number A74 (hundreds digit A, tens 7, ones 4) times a single digit B equals 6992. Find A and B.

Givens

A74 x B = 6992.
A74 has tens digit 7 and ones digit 4; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A74 is three digits).
B is a single digit (1 through 9).

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 4 times B must end in 2, which only a couple of digits do. Test each candidate B by dividing 6992 to see if it gives a clean A74 form.

Execute

#2 Make a Systematic List 3.OA.C.7

The product ends in 2, and the ones digit of A74 is 4, so 4 x B must end in 2. Checking 4 x 1, 4 x 2, ...: only 4 x 3 = 12 and 4 x 8 = 32 end in 2. So B is 3 or 8.

4 \times 3 = 12,\quad 4 \times 8 = 32

The ones digit of a product depends only on the ones digits of the factors.

#6 Guess and Check 3.OA.A.4

If B = 3, then A74 = 6992 / 3, which is not a whole number, so B is not 3. If B = 8, then A74 = 6992 / 8 = 874, which fits the pattern A74 with A = 8.

6992 \div 8 = 874

Dividing the product by B should rebuild the original number with 7 in the tens and 4 in the ones place.

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

Check 874 x 8: 800 x 8 = 6400, 70 x 8 = 560, 4 x 8 = 32, total 6992. So A = 8 and B = 8.

874 \times 8 = 6400 + 560 + 32 = 6992

Multiplying back must return the original 6992.

Answer: A = 8, B = 8

Review

874 x 8 = 6992 matches exactly, the tens digit is 7 and ones digit is 4 as required, so the solution fits all conditions.

Estimate B: 6992 is near 7000, and 7000 / 874 is about 8, pointing straight to B = 8, then A74 = 874 gives A = 8.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking 874 x 8.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Recovering A74 by dividing 6992 by the candidate B.

💡 Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!