Depth · one concept from first principles · step 2 of 4

Place digit cards for largest or smallest product

From the four number cards $2$ , $4$ , $5$ , and $6$ , choose $3$ of them and use each chosen card once to build a single multiplication of the form (two-digit number) $\times$ (one-digit number). Find the product when it is as large as possible, and the product when it is as small as possible.

Base-ten numbersOperations

Your answer

How to solve

Strategy Guess and Check — There are only a handful of sensible arrangements, so make a short systematic list of the strong candidates and check each product. For the largest product, big digits in high place values matter; for the smallest, small digits do. Testing the top few candidates each way pins down the extremes with certainty.

1STEP 1

Aim for the largest product

To make the product large, the multiplier should be large and the two-digit number large. Test 65 × 4, 64 × 5, and 54 × 6.

65 × 4 = 260, 64 × 5 = 320, 54 × 6 = 324

2STEP 2

Pick the maximum

Among the candidates, 54 times 6 gives the biggest product, so the largest possible product is 324.

54 × 6 = 324

3STEP 3

Aim for the smallest product

To make the product small, the multiplier should be small (the card 2) and the two-digit number small. Test 45 × 2 and 46 × 2.

45 × 2 = 90, 46 × 2 = 92

4STEP 4

Pick the minimum

The smallest of these is 45 times 2, so the smallest possible product is 90.

45 × 2 = 90

Answer

Largest product 324 (54 × 6); smallest product 90 (45 × 2)

Both products use exactly three of the four cards once. 324 is near the top of what these cards allow (under 70 times 6 = 420 but using only valid cards), and 90 is small as expected with the 2 as multiplier. A quick scan of other arrangements (65 times 4 = 260, 56 times 4 = 224) stays between 90 and 324, confirming the extremes.

Takeaway

Big digit as the multiplier for the most, small digits for the least: Grade 3 place-value thinking!

Aim for the largest product
Pick the maximum
Aim for the smallest product
Pick the minimum

Where next?

Level upsuggested

Smaller divisor makes a bigger quotient Build largest and smallest mixed numbers from cards

Another one like thissuggested

10 practice problems

Same look, different logicsuggested

Recover missing digits from column carries

Step back and solidifysuggested

Big digits in high places Count numbers built from digit cards

8 reasoning problems grow out of this type — the bridge toward competition →

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 10 problems

The same shape shows up later on the AMC, as Branching State Enumeration, Casework by Position / Vertex Type. Why the shapes repeat →