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CAT 2020 Slot 1 — QA Question 12

FactorsEasy

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

Answer & solution

Answer: 21

Solution

Easy

The digit product must be one of 3,4,5,63,4,5,6. For each target value, list the unordered digit triples that multiply to it (digits from 11–99), then count how many distinct 3-digit numbers each triple produces. A 3-digit number has no leading-zero issue here because a 00 digit would make the product 00.

1

Translate the condition. If the digits are p,q,rp,q,r, the product must lie strictly between 22 and 77.

2<p×q×r<7⇒ p×q×r∈{3,4,5,6}(integer products)\begin{aligned} &2 < p\times q\times r < 7\\ &\Rightarrow\ p\times q\times r \in \{3,4,5,6\} \quad\text{(integer products)} \end{aligned}
2

Products 33 and 55 (primes). A prime product forces the digits to be that prime with two 11s, i.e. digit-set {1,1,p}\{1,1,p\}. The single distinct digit can sit in any of 33 positions.

product 3: {1,1,3}⇒3 numbers(113, 131, 311)product 5: {1,1,5}⇒3 numbers(115, 151, 511)\begin{aligned} &\text{product }3:\ \{1,1,3\}\Rightarrow 3\ \text{numbers}\quad\text{(113, 131, 311)}\\ &\text{product }5:\ \{1,1,5\}\Rightarrow 3\ \text{numbers}\quad\text{(115, 151, 511)} \end{aligned}
3

Product 44. Write 44 as a product of three digits in all ways, then count arrangements of each multiset.

{1,1,4}⇒3 arrangements{1,2,2}⇒3 arrangements⇒ 3+3=6 numbers(114,141,411,122,212,221)\begin{aligned} &\{1,1,4\}\Rightarrow 3\ \text{arrangements}\\ &\{1,2,2\}\Rightarrow 3\ \text{arrangements}\\ &\Rightarrow\ 3+3 = 6\ \text{numbers} \quad\text{(114,141,411,122,212,221)} \end{aligned}
4

Product 66. Same idea.

{1,1,6}⇒3 arrangements{1,2,3}⇒3!=6 arrangements⇒ 3+6=9 numbers\begin{aligned} &\{1,1,6\}\Rightarrow 3\ \text{arrangements}\\ &\{1,2,3\}\Rightarrow 3! = 6\ \text{arrangements}\\ &\Rightarrow\ 3+6 = 9\ \text{numbers} \end{aligned}
5

Add them up.

3+3+6+9=21(products 3,5,4,6)\begin{aligned} &3+3+6+9 = 21 \quad\text{(products }3,5,4,6\text{)} \end{aligned}
2121
CAT 2020 Slot 1 QA Q12: How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7? — Solution | TheCATExam