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Factors & Multiples – A simple way of understanding them

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Factors & Multiples – A simple way of understanding them  [#permalink]

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New post 21 May 2019, 18:41
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It’s not hard to figure out that, if you want to do well in Math questions, you need to be well versed with the most fundamental aspect of Math – numbers. After all, numbers are the building blocks of Math, just like alphabets are the building blocks of a language. As such, being well conversant with all aspects of a topic like Numbers is of paramount importance, if you want to do well on questions related to Numbers and also on Math problems in general.

One of those aspects of Numbers, which is tested very frequently in GMAT questions, is that of Factors and Multiples. Although these are simple concepts by themselves, a lot of students always have a confusion and sometimes end up using one in the place of the other.

As tutors, a common mistake we see students making, is in finding out the HCF and LCM. They end up finding out the LCM when HCF is asked. This problem only gets accentuated when the question is a Word problem where a student has to identify whether it is the HCF that has to be computed or the LCM.

We also see that, in questions where a number is said to be a factor of another, students find it hard to form equations linking the two numbers. This will obviously have a major say in the solution, especially if the question is a Data Sufficiency question.

So, we thought that this was an ideal topic for us to throw our weight around and let you folks know that there is an easy way of understanding factors and multiples.

Let us start with an example. If 37 is divided by 5, we know that the quotient is 7 and the remainder is 2. In this case, 37 is the dividend and 5 is the divisor. A point to note is, we used the term divisor and not factor. Why?

Let’s take another example. If 35 is divided by 5, the quotient is again 7, but, this time the remainder is 0. In this case, all of you will unanimously agree that 5 is a factor of 35. This is because, 5 was able to divide 35 completely, without leaving any remainder. This is also the reason why you cannot term 5 as a factor of 37.

So, let’s look at a simple definition of the term factor:
“A factor is a number that can divide another number completely”. Easy to remember and understand, right?
Also note that the word factor will always be related to division, as you saw in the examples above. So, a thumb rule you can follow, when it comes to analyzing word problems, is that if you have to divide/distribute/split a given number into smaller parts, you have to consider factors of that number.
Technically, if x is a factor of y, then y/x = k or y = kx, where k is an integer.

Let us now look at the term multiple.
In one of the examples above, we termed 5 as the factor of 35. The flip side is that 35 is a multiple of 5.
Why is this? This is because the number 35 can be found in the multiplication table of 5. In that case, 35 should also be a multiple of 7, you say?
Absolutely, 35 is a multiple of 7 also. In fact it is a common multiple of 5 and 7. More on common multiples later.
Note: 7 is a factor of 35, for the same reasons that 5 is.

From the above examples, we can conclude that:
“A multiple is a number that can be completely divided by another number OR
A multiple is a number that can be found in the multiplication table of another number”
You can also infer that the term multiple is used analogously with multiplication. So, in situations where you are trying to find out numbers which could be in the multiplication table of a given number/s, you are dealing with multiples of that number.
Technically, if x is a multiple of y, then x= y*k, where k is an integer.

If you observed something interesting, you will see that the factor of a number will always be smaller or at most equal to the number itself. Similarly, a multiple of a number will be equal or bigger than the number. This is another trick that you can use to make your life simple when analyzing word problems on HCF and LCM and in general also, while dealing with factors and multiples.

Lastly, let us look at a real life example which will make understanding factors and multiples, really easy.
Let us say you had $100 with you and you were trying to get change for this amount. Ask yourself,
Can I get change for $100 in terms of $1 bills? The answer would be YES, because you can divide $100 into 100 equal parts of $1 each.
Can I get change for $100 in terms of $2 bills? The answer would again be YES, because of the similar reasons.
Can I get change for $100 in terms of $5, $10, $20 and $50 bills? The answer would be YES.
This is because, in each of the cases, the smaller denominations represent factors of the number 100. That is why you were able to divide $100 into so many equal parts.
But, can you get change for $100 in terms of $500 bills (I know it sounds weird, but just a hypothetical case to extend the example)? You would surely say NO. That is because, 500 is not a factor of$100.
I hope all of you have a clearer and better understanding of the term factor, now.

Similarly, let us say you had a bunch of $100 bills with you. Will you be able to lend $200 dollars to someone? Yes, that’s possible. How about $700? That’s possible too.
How about $950? That’s not possible because 950 is not a multiple of 100.

In this example also, you can observe that the factors of 100 were all smaller than 100, whereas the multiples of 100 were bigger than 100.
This must have certainly cleared the air on the difference between a factor and a multiple. It would also have given you an easy way of finding out either of them.

In our next part of this post, tomorrow, we will extend this discussion to HCF and LCM. We hope that you liked this post. If you did, go ahead and give us Kudos to motivate us to write more such articles on GMAT Quant.

See you in the next part of this post!
Thank you!
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Factors & Multiples – A simple way of understanding them   [#permalink] 21 May 2019, 18:41
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