Problem
Exact division means remainder zero
The two-digit number 7-followed-by-a-blank must divide exactly by 4. We need every digit the blank could be so that the number is a multiple of 4.
Operations
Your answer
Watch out It's tempting to think any even ones digit works — but divisibility by 4 depends on the whole two-digit number, not just evenness.
How to solve
Strategy Make a Systematic List — There are only ten numbers from 70 to 79, so we can list the multiples of 4 in that range and read off which ones-digits work.
1STEP 1
List multiples of 4 near 70
The multiples of 4 are 4 times 18 equals 72 and 4 times 19 equals 76. The next one, 4 times 20, is 80, which is past the 70s.
4 × 18 = 72, 4 × 19 = 76
Listing the multiples of 4 inside 70 to 79 catches every number that divides evenly.
3.OA.C.7Make A Systematic ListAmong the numbers 70 through 79, exactly 72 and 76 divide by 4 with no remainder.
Why?
A number divides by 4 with no remainder exactly when it is built from a whole number of 4s, so the numbers we want are the multiples of 4.
Why?
Sharing a number into equal groups of 4 with nothing left over is the reverse of building that number from a whole count of 4s, so 'divides by 4 exactly' and 'is a multiple of 4' pick out the same numbers.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The multiples of 4 land every 4 steps — 68, then 72, then 76, then 80 — so inside 70 through 79 only 72 and 76 fit.
Why?
Each next multiple of 4 is the one before it plus one more group of 4, so neighboring multiples always sit exactly 4 apart.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Read off the blank digits
Both 72 and 76 start with 7, so the blank can be 2 or 6. 70, 74, 78 leave remainders, so they don't work.
72 ÷ 4 = 18, 76 ÷ 4 = 19
Each valid number gives a whole quotient, confirming the digit makes a clean division.
3.OA.B.6Guess And CheckAnswer
2 or 6
72 and 76 are the only multiples of 4 between 70 and 79 (multiples of 4 are spaced 4 apart, so a ten-wide window holds two or three of them). Both give whole quotients, so the answers are correct.
Takeaway
This only needs Grade 3 multiplication facts: list the multiples of 4 and see which land in the 70s!
- List multiples of 4 near 70
- Read off the blank digits
Where next?
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