Problem
Largest and smallest product
Use the cards 2, 3, 5, 8 once each to fill the blanks of each multiplication. The product must hit the stated target exactly. Decide which digit goes where in all four parts.
Reasoning · Grade 3-2 Multiplication Patterns
Your answer
Watch out Testing every arrangement is slow — the product's ones digit cuts the candidates at once.
How to solve
Strategy Guess and Check — There are only a handful of ways to split four digits into the required shapes, so I can guess promising arrangements and check. Looking at the ones digit of the target (the units) sharply narrows which digit can be the one-digit factor or the units digit, so each part needs only a couple of checks.
1STEP 1
(1) two-digit × two-digit = 4346
4346 ends in 6 from 3 × 2, giving 53 × 82.
53 × 82 = 4346
A Grade 3 learner can test one promising pairing and multiply to confirm, using the ones-digit clue to avoid wild guessing.
3.NBT.A.2Guess And Check2STEP 2
(2) three-digit × one-digit = 4256
4256 ÷ 8 is 532, so 532 × 8.
532 × 8 = 4256
Checking the units: 8 times a number ending in 2 ends in 6, matching 4256, which points straight to 8 as the single factor.
3.NBT.A.3Analyze The UnitsBecause 8 times a number ending in 2 ends in 6, the single-digit factor of 4256 has to be 8.
Why?
The ones digit of a product depends only on the ones digits being multiplied, because every higher place carries a whole ten upward.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Testing each single-digit card against that ones-digit rule throws out every card but one, so no further searching is needed.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
(3) two-digit × two-digit = 950
950 ends in 0, needing a 5 against an even: 25 × 38.
25 × 38 = 950
A product ending in 0 needs a 5 and an even number multiplied, so the units digit immediately tells me how to place the 5.
3.NBT.A.2Analyze The Units4STEP 4
(4) three-digit × one-digit = 716
716 ÷ 2 is 358, so 358 × 2.
358 × 2 = 716
Because 716 is just over 700, the single factor must be the small card 2, and dividing back gives the three-digit number directly.
3.NBT.A.3Guess And CheckAnswer
53 × 82 = 4346, 532 × 8 = 4256, 25 × 38 = 950, 358 × 2 = 716
Each arrangement uses 2, 3, 5, 8 exactly once and the multiplication checks out: 53×82=4346, 532×8=4256, 25×38=950, 358×2=716. The ones digits also match each target (6, 6, 0, 6), confirming the placements.
Takeaway
Peek at the last digit of the answer first — it tells you exactly where the 5 or the even card has to go, so you barely have to guess!
- (1) two-digit × two-digit = 4346
- (2) three-digit × one-digit = 4256
- (3) two-digit × two-digit = 950
- (4) three-digit × one-digit = 716