Problem
Multiplication synthesis challenge
In the column multiplication, AB × A gives the partial product 16A. Shifting the two-digit partial product CC and adding makes DAB. A, B, C and D are different single digits. Find all four.
Reasoning · Grade 3-1 Multiplication
Your answer
Watch out It is easy to add CC flush, but as the second partial product it is shifted one place.
How to solve
Strategy Guess and Check — The partial product 16A ends in A and equals AB x A, which forces A almost immediately; once A is known the column addition 16A + CC = DAB determines B, C, and D one place at a time. A short guess-and-check on A pins everything down.
1STEP 1
Pin down A from the partial product
A partial product in the 160s ending in A forces A = 4, AB = 41.
41 × 4 = 164 = 164 → A = 4, B = 1
A 3rd grader can just try the multiplication facts that give a product in the 160s and read the ones digit.
3.OA.C.7Guess And Check2STEP 2
Add the ones column to get B
The total's ones digit matches AB's, so B = 1.
ones: 4 + C ≡ B (mod 10)
Adding the right-hand column first is exactly how kids add multi-digit numbers, carrying when needed.
3.NBT.A.2Work BackwardsAdding the ones column first gives B, and any ten it makes moves up into the next column.
Why?
Each column handles one size of bundle and can pass only whole bundles upward, so the ones column can be settled before the tens is touched.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The full total is what all the column contributions come to together, so nothing is lost when a ten leaves one column and joins the next.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Find C from the tens column
4 + C must end in 1, so C = 7.
4 + 7 = 11 (carry 1); 6 + 7 + 1 = 14 → tens digit 4 = A
Listing the two leftover digits that add to a number ending in 1 leaves only C = 7.
3.NBT.A.2Make A Systematic List4STEP 4
Find D from the hundreds column
With the carry the hundreds make 2, so D = 2.
164 + 77 = 241 = 2 4 1 = DAB
The final carry plus the leading 1 simply gives the hundreds digit of the total.
3.OA.A.4Guess And CheckAnswer
A = 4, B = 1, C = 7, D = 2 (41 × 4 = 164, 164 + 77 = 241)
Check every line: 41 x 4 = 164 = 16A with A = 4; CC = 77; 164 + 77 = 241 = DAB with D = 2, A = 4, B = 1. The digits 4, 1, 7, 2 are all distinct, so the different-letter rule holds and the whole layout is consistent.
Takeaway
Work one column at a time, carrying as you go - the same way you already add and multiply in Grade 3!
- Pin down A from the partial product
- Add the ones column to get B
- Find C from the tens column
- Find D from the hundreds column