# How many 3-digit numbers are there... | Quantitative Aptitude - CAT Number Series

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The question below is from previous year CAT question from CAT 2020 exam comes from CAT Number Series: How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

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## CAT 2020 - Slot 1 - Quantitative Aptitude - Question 1 - How many 3-digit numbers are there

Q.1: How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

Let the digits of the 3-digit number be p, q, & r.

2 < p × q × r < 7

Therefore, p × q × r can take the values 3, 4, 5, or 6.

Since they are prime, they can’t be split, and hence if one of p,q or r is 3, the remaining two should be 1.

So, the possible combinations are

1, 1, 3

1, 3, 1

3, 1, 1

1, 1, 5

1, 5, 1

5, 1, 1

4 can be split as 2 × 2. Therefore, the possible combinations of p, q, r are

1, 1, 4

1, 4, 1

4, 1, 1

1, 2, 2

2, 1, 2

2, 2, 1

6 can be split as 3 × 2. Therefore, the possible combinations of p, q, r are

1, 1, 6

1, 6, 1

6, 1, 1

1, 2, 3 will also yield a product of 6. We can 3! = 6 combinations of p, q, r with 1, 2, 3

1, 2, 3

1, 3, 2

2, 1, 3

2, 3, 1

3, 1, 2

3, 2, 1

Therefore, the total number of possibilities are 3 + 3 + 3 + 3 + 3 + 6 = 21

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