Question & Answer
QUESTION

The smallest 4 digit positive number exactly divisible by 12, 15, 20 and 35 is which of the following:
(a) 1000
(b) 1160
(c) 1260
(d) None of these

ANSWER Verified Verified
Hint: Start by finding the smallest 4 digit positive number which is 1000. Then find the LCM of 12, 15, 20 and 35. Then write the multiplication table of the LCM. The first number in the multiplication table which is greater than or equal to 1000 is the final answer.

Complete step-by-step answer:

In this question, we need to find the smallest 4 digit positive number exactly divisible by 12, 15, 20 and 35.

Let us first see the smallest 4 digit number. Smallest 4 digit positive number = 1000
Now, we need to find the LCM of 12, 15, 20 and 35.

2∣12, 15, 20, 35​
2∣6, 15, 20, 35 ​
3∣3, 15, 5, 35 ​
5∣1, 5, 5, 35 ​
7∣1, 1, 1, 7 ​
  |1, 1, 1, 1

So, the LCM of 12, 15, 20 and 35 is 2 $\times $ 2 $\times $ 3 $\times $ 5 $\times $ 7 = 420

So, 420 is the smallest positive number, which is divisible by 12, 15, 20 and 35. But it is not a 4 digit number. We know that if 420 is divisible by 12, 15, 20 and 35. Then the multiples of 420 will also be divisible by 12, 15, 20 and 35. So, we will now see the multiples of 420.

We will now see the multiplication table of 420 and the first number which is greater than or equal to 1000 will be our answer.

420 $\times $ 1 = 420

420 $\times $ 2 = 840

420 $\times $ 3 = 1260

So, 1260 is the first multiple of 420 which is greater than 1000. So, 1260 is the smallest 4 digit positive number exactly divisible by 12, 15, 20 and 35.

Hence, option (c) is correct.

Note: In this question, we have to find the smallest positive integer, which is divisible by 12, 15, 20 and 35. But if we were asked the smallest integer, which is divisible by 12, 15, 20 and 35 then the answer would have been -9660. So, it is important to carefully read and understand the question to avoid any mistakes.