Question of The Day: Factors & Multiples

How many positive integers up to 1000

Q: How many positive integers up to 1000 exist that are divisible by 2 or 5?
 

Quick Solution for Experts:

No. of positive integers up to 1000 that are divisible by 2 = 1000/2 = 500
No. of positive integers up to 1000 that are divisible by 5 = 1000/5 = 200

However, all multiples of 10 up to 1000 are counted twice,

Important Note: 10 (being the LCM of 2 and 5) and all multiples of 10 are counted twice because all multiples of 10 are multiples of both 2 as well as 5. Thus, we need to find number of these integers that are counted twice, and then subtract these from 700.

No of positive integers up to 1000 that are divisible by 10 = 1000/10 = 100

Finally, as we know,
Multiples of 2 or Multiples of 5 = Multiples of 2 + Multiples of 5 – Multiples of both 2 and 5 (i.e. multiple of 10)

                                                        = 500 + 200 – 100
                                                        = 600 Answer!

 

Explanatory solution for beginners:

Positive integers that are divisible by 2 are basically multiples of 2. Similarly, those positive integers that are divisible by 5 are, in fact, multiple of 5.

In general words, number of integers divisible by an integer x basically means all multiples of x.

How many multiples of 2 exists in the first 1000 positive integers?

In order to learn this, let’s consider a small scale scenario and see how it goes to learn the technique:
(The below concept is a part of our complete online course)

Finding multiple of 7 in the first 20 positive integers:
Multiple of 7:   7, 14, 21, 28, 35,

What will happen if we divide 20 by 7, i.e.
= 20/7 = 2.85… (Two Point Something…)
This integer part gives the required result. Hence, there are two multiples of 7 in the first 20 positive integers.

So, no. of multiples of 2 up to 1000 positive integers = 1000/2 = 500
Similarly, multiples of 5 up to 1000 positive integers = 1000/5 = 200
Totaling: 500 + 200 = 700

However,
Multiple of 2:   2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ……, 1000
Multiple of 5:   5, 10, 15, 20, ……, 1000

Notice that some integers comes in multiples of both 2 and 5. These are actually counted twice. In order to know these, you need to answer:

What is the LCM (Least Common Multiple) of 2 and 5?
LCM and all of its multiples are included in multiples of 2 as well as in multiples of 5. So, in the total (500 + 200 = 700), these integers are counted twice. And we need to subtract them once for getting the correct answer.

Thus, we need to find how many integers are there that are counted twice. For this purpose, we should find multiples of 10 up to 1000 by using the same method as you have learned for multiples of 2 and that of 5.

Multiples of 10 up to 1000 = 1000/10 = 100

Now that you have found no. of integers that were counted twice. So, let’s subtract it once from 700 to find correct answer.

Multiples of 2 or Multiples of 5 up to 1000 = 700 – 100 = 600 Answer!

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