Problem
Knowing the multiple aids factoring into products
The four-digit number 284 followed by one missing digit must be a multiple of 6. I need to count how many different digits (0 through 9) can fill that blank.
Operations
Your answer
How to solve
Strategy Make a Systematic List — Being a multiple of 6 means being a multiple of 2 AND a multiple of 3. So I first list only the digits that keep the number even (the multiple-of-2 rule), then check each one against the multiple-of-3 rule (the digit sum). Listing the few candidates and checking them is faster and surer than dividing many numbers.
1STEP 1
Split 6 into 2 and 3
A multiple of 6 is a multiple of 2 and of 3, so the last digit settles one rule and the digit sum the other.
6 = 2 × 3
Breaking 6 into its factors 2 and 3 turns one hard divisibility test into two easy ones a fourth grader already knows.
4.OA.B.4Make A Systematic ListA number is a multiple of 6 exactly when it is a multiple of 2 and a multiple of 3.
Why?
6 is a 2 and a 3 multiplied together, so anything built out of whole 6s is also built out of whole 2s and out of whole 3s.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The reverse holds as well: 2 and 3 have no factor in common, so a number carrying a whole 2 and a whole 3 must carry a whole 6.
Why?
The factors making up the number can be regrouped in any order, so a 2 and a 3 can always be pushed together into a single 6.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
2STEP 2
Keep only the even endings
The even rule leaves only 0, 2, 4, 6, 8 to test instead of all ten digits.
■ ∈ {0, 2, 4, 6, 8}
Applying the easy even-number rule first shrinks the list, so there is far less to test.
4.OA.C.5Make A Systematic List3STEP 3
Check each even ending against the multiple-of-3 rule
The fixed digits add to 14, and only 14 + 4 = 18 is a multiple of 3, so the box is 4.
2+8+4=14; 14+0=14, 14+2=16, 14+4=18, 14+6=20, 14+8=22
Testing five small sums against the multiple-of-3 rule is quick and leaves no doubt about which digit fits.
4.OA.B.4Guess And Check4STEP 4
Count the digits that work
Only ■ = 4 passes both rules, giving the number 2844. So exactly one digit can fill the blank.
2844 ÷ 6 = 474
Dividing 2844 by 6 with no remainder confirms the one survivor is really a multiple of 6.
4.NBT.B.6Make A Systematic ListAnswer
1 digit
There can be at most five even endings, so an answer of 1 is in range. Checking directly: 2844 ÷ 6 = 474 with no remainder, while the next even endings 2840, 2842, 2846, 2848 each leave a remainder, confirming 4 is the only digit that works.
Takeaway
To find multiples of 6, use the easy 2-rule and 3-rule together instead of dividing big numbers.
- Split 6 into 2 and 3
- Keep only the even endings
- Check each even ending against the multiple-of-3 rule
- Count the digits that work
Where next?
Another one like thissuggested
Step back and solidifysuggested