Narrow candidates by divisibility conditions

3.OA.B.63.OA.C.7 · take · grade 3

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

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $4$ and $6$ .
It is less than $30$ .
The sum of its tens digit and its ones digit is $6$ .

Show solution

Understand

We need a two-digit number that is divisible by both 4 and 6, is less than 30, and whose tens digit and ones digit add up to 6.

Givens

The number is divisible by 4 and also by 6.
The number is less than 30.
The tens digit plus the ones digit equals 6.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 30.
Divisible by both 4 and 6 means divisible by 12.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 4 and 6, i.e. by 12) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 4 and 6 is a multiple of 12. List the multiples of 12: 12, 24, 36, ... .

\text{multiples of }12: 12, 24, 36, \ldots

Grade 3 times-tables: being in both the 4 and 6 tables means being in the 12 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 12 under 30: that leaves 12 and 24 (36 is too big).

12 < 30, \quad 24 < 30, \quad 36 \not< 30

Grade 3: only the multiples of 12 that fit under 30 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums: 12 has 1 + 2 = 3 (no), while 24 has 2 + 4 = 6 (yes). Only 24 has a digit sum of 6.

1 + 2 = 3, \quad 2 + 4 = 6

Grade 3 place value and addition: adding the two digits of 24 gives exactly 6.

Answer: 24

Review

Verify 24 against all conditions: 24 div 4 = 6 and 24 div 6 = 4 (divisible by both), 24 < 30 (true), and 2 + 4 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 30 whose digits add to 6 (15, 24) and then keep only the one divisible by both 4 and 6, which is 24.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 12 from the 4 and 6 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 4 and 6 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 4 and 6 means a multiple of 12, and the only one under 30 with digits adding to 6 is 24!