Problem
Narrow candidates by divisibility conditions
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.
Operations
Your answer
Watch out Don't stop once you've narrowed it to 12 or 24 — you still must check the digit-sum clue before picking one.
How to solve
Strategy Make a Systematic List — 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.
1STEP 1
Combine the two divisibility clues
A number divisible by both 4 and 6 is a multiple of 12. List the multiples of 12: 12, 24, 36, ... .
multiples of 12: 12, 24, 36, ...
Grade 3 times-tables: being in both the 4 and 6 tables means being in the 12 table.
3.OA.C.7Make A Systematic ListA number divisible by both 4 and 6 must be a multiple of 12, so the only numbers to check are 12, 24, 36, and so on.
Why?
12 is itself divisible by both 4 and 6, so it is the first number that can satisfy the two divisibility clues at once.
Why?
Counting 4 three times reaches 12 (4, 8, 12), so 12 is 3 equal groups of 4 and 4 divides it evenly.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Counting 6 two times reaches 12 (6, 12), so 12 is 2 equal groups of 6 and 6 divides it evenly.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The numbers that appear in both the 4-times table and the 6-times table are exactly the 12-times table, so any number divisible by both 4 and 6 lands on a multiple of 12.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Apply the 'less than 30' clue
Keep only the two-digit multiples of 12 under 30: that leaves 12 and 24 (36 is too big).
12 and 24 are less than 30, but 36 is not
Grade 3: only the multiples of 12 that fit under 30 stay in the running.
3.OA.B.6Eliminate Possibilities3STEP 3
Apply the digit-sum clue
Digit sums: 12 gives 1+2=3, but 24 gives 2+4=6 — only 24 fits.
1 + 2 = 3, 2 + 4 = 6
Grade 3 place value and addition: adding the two digits of 24 gives exactly 6.
3.OA.A.3Eliminate PossibilitiesAnswer
24
Verify 24 against all conditions: 24 div 4 = 6 and 24 div 6 = 4 (divisible by both), 24 is less than 30 (true), and 2 + 4 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.
Takeaway
Divisible by 4 and 6 means a multiple of 12, and the only one under 30 with digits adding to 6 is 24!
- Combine the two divisibility clues
- Apply the 'less than 30' clue
- Apply the digit-sum clue
Where next?
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