GATE CSE 2014 | Set 2 | Question: 49

Question

The number of distinct positive integral factors of $2014$ is _____________

Comments

@chirudeepnamini

Click on the above comment reference.

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@Lakshman Patel RJIT thankyou:)

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for the easier understanding the difference between Integral factors Vs prime factors follow this link and the following chat of sachin sir to get better understanding:

 

Answer 1

First do prime factorization of $2014 - 2^{1} \times 19^{1} \times 53^{1}$

Now to get a factor of $2014,$ we can choose any combination of the prime factors including $0.$ i.e; $2^{0}$ and $2^{1}$ are possible and similarly for other prime factors also, there are $2$ possibilities. So, the total number of positive integral factors

$= 2 \times 2 \times 2 = 8$

(When all the powers of prime factors are $0,$ we get $1$ and when all the powers are maximum, we get the given number.)

Comments

But $1007=6*168 - 1 (6n-1)$ and is not a prime,so dont use the $6n+1$ or $6n-1$ method to judge primality.It is better to divide by factors $<\sqrt{1007}$.

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if power of 2 was like 2²  then we could have three combinations as 2^0 2^1 and 2^2. Right?

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53 cannot be expressed as 6n+1. Also, 1007 can be expressed as 6n-1 from the above comment.

So just go with the method given above which is trying to divide by prime numbers$\leq \left \lfloor \sqrt{1007} \right \rfloor$

Answer 2

2014 = 2 x 19 x 53

total  positive integral factor=(2x2x2)=8

Answer 3

the factors are 1, 2, 19, 38, 53, 106, 1007 and 2014. So the total makes it 8 number of integral factors.

Comments

@rishu_darkshadow

No. It is simple prime factorization

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okzz..

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good

Answer 4

2014=2^1*19^1*53^1--->(1+1)(1+1)(1+1)

using theorem = 8.