TANCET - XAT Math Question : Arithmetic Progression

The question given below is a sample practice problem in Arithmetic Progression.

Question

How many 2-digit positive integers are divisible by 4 or 9?

1. 32
2. 22
3. 30
4. 34
5. 10

Correct Answer - 30. Choice (3)

Explanatory Answer

Number of 2-digit positive integers divisible by 4

The smallest 2-digit positive integer divisible by 4 is 12. The largest 2-digit positive integer divisible by 4 is 96.

All the 2-digit positive integers are terms of an Arithmetic progression with 12 being the first term and 96 being the last term.

The common difference is 4.

The nth term a_n = a₁ + (n - 1)d, where a₁ is the first term, 'n' number of terms and 'd' the common difference.

So, 96 = 12 + (n - 1) * 4

Or 84 = (n - 1) * 4

Or (n - 1) = 21

Hence, n = 22.

i.e., there are 22 2-digit positive integers that are divisible by 4.

Number of 2-digit positive integers divisible by 9

The smallest 2-digit positive integer divisible by 9 is 18. The largest 2-digit positive integer divisible by 9 is 99.

All the 2-digit positive integers are terms of an Arithmetic progression with 18 being the first term and 99 being the last term.

The common difference is 9.

The nth term a_n = a₁ + (n - 1)d, where a₁ is the first term, 'n' number of terms and 'd' the common difference.

So, 99 = 18 + (n - 1) * 9

Or 81 = (n - 1) * 9

Or (n - 1) = 9

Hence, n = 10.

i.e., there are 10 2-digit positive integers that are divisible by 9.

Removing double count of numbers divisible by 4 and 9

Numbers such as 36 and 72 are multiples of both 4 and 9 and have therefore been counted in both the groups.

There are 2 such numbers.

Hence, number of 2-digit positive integers divisible by 4 or 9
= Number of 2-digit positive integers divisible by 4 + Number of 2-digit positive integers divisible by 4 - Number of 2-digit positive integers divisible by 4 and 9

= 22 + 10 - 2 = 30

Correct answer choice (3).

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