Context: 6.5 The Largest Product

Q. Fill the digits 2, 3, and 5 in $\square\square \times \square$ , using each digit once. What is the largest product possible?

Question

Context: 6.5 The Largest Product Q. Fill the digits 2, 3, and 5 in , using each digit once. What is the largest product possible?

Homi · Accepted Answer

Understand the Question

We are given the digits $2$ , $3$ , and $5$ , each to be used exactly once in the form $\square\square \times \square$ .
To find the largest product, we test each digit as the single-digit multiplier while forming the largest possible two-digit number with the remaining two digits (placing the larger remaining digit in the tens place).
Finally, we compare all resulting products to identify the maximum.

Step 1 · Calculate Products for Each Multiplier

To maximize the product for each choice of single digit, place the larger of the remaining two digits in the tens place:

Case 1: Single digit is $5$ Remaining digits are $2$ and $3$ . The largest two-digit number is $32$ . $32 \times 5 = 160$

Case 2: Single digit is $3$ Remaining digits are $2$ and $5$ . The largest two-digit number is $52$ . $52 \times 3 = 156$

Case 3: Single digit is $2$ Remaining digits are $3$ and $5$ . The largest two-digit number is $53$ . $53 \times 2 = 106$

Step 2 · Compare Products

Comparing the values obtained from all cases: $106 < 156 < 160$

The largest possible product is $160$ .

Answer

$160$

Common Mistakes

Assuming the Largest Two-Digit Number Gives the Largest Product: It is a common mistake to make the two-digit number as large as possible ( $52 \times 3 = 156$ ) instead of using the largest digit as the multiplier ( $32 \times 5 = 160$ ), which scales the entire number more.
Incorrect Digit Order in Two-Digit Number: Placing the smaller digit in the tens place (e.g., $23 \times 5 = 115$ instead of $32 \times 5 = 160$ ) reduces the product significantly.

Question 12