GATE CSE 2015 | Set 2 | Question: 9

Question

The number of divisors of $2100$ is ____.

Comments

https://gateoverflow.in/2015/gate2014-2-49

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HER THIS MIGHT HELP ARE REALLY HELPFUL THANKS

Answer 1

Answer: 36

$2100 = 7\times 3\times 2^2 \times 5^2$

Hence, total number of factors will be $= (1+1)\times (1+1)\times (2+1)\times (2+1) \\= 2 \times 2\times 3 \times 3 \\= 36$,

because any factor is obtained by multiplying the prime factors zero or more times. (one extra for zero)

Comments

          2100
           /   \
        2    1050
                /  \
             2    525
                    /  \
                  3    175
                         /  \
                        5   35
                              / \
                            5   7
                      
2² 3¹ 5²  7¹   => (2+1)*(1+1)(2+1)*(1+1) => 3*2*3*2 = 36
 

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@Dq Our objective Should here to do prime factorization...can be done in many ways

2100=21*100=3*7*$5^2$*$2^2$

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Since in the question it's not mentioned about positive or negative divisor, then shouldn't we find both negative and positive divisors, which are respectively 36 and 36, which sums up to 72 as total number of divisors?

Answer 2

 

$2100 = 7 * 3 * 2^{2} * 5^{2}$

To calculate the total number of divisors, take each power in the prime factorization obtained, and:

Step 1: Increment the power.

$(1+1) (1+1) (2+1) (2+1)$

Step 2: Calculate the product of it:

$(1+1) * (1+1) * (2+1)* (2+1)$

Answer: 36

Answer 3

Ans  - 36

Detailed Explanation to get clear understanding

Start with 2100:

• 2100 ÷ 2 = 1050

• 1050 ÷ 2 = 525 → So 2²

• 525 ÷ 3 = 175 → So 3¹

• 175 ÷ 5 = 35

• 35 ÷ 5 = 7 → So 5²

• 7 is a prime → 7¹

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Final Prime Factorization of 2100:

2100 = 2² × 3 × 5² × 7

Using the divisor formula:
d(n) = (a₁ + 1) × (a₂ + 1) × ... × (aₖ + 1)

d(2100) = (2+1)(1+1)(2+1)(1+1)=3⋅2⋅3⋅2=36 divisors

Answer 4

Prime factors for 2100 is 2²X5²X3X7

 

So, we got that possibilities for factors like

 

For 2 it 2⁰, 2¹ or 2² so there are 3 choices for it

 

For 5 it is 5⁰, 5¹ or 5² so there are 3 choices for it

 

For 3 it is 3⁰ or 3¹ so there are 2 choices for it

 

For 7 it is 7⁰ or 7¹ so there are 2 choices for it

 

Hence multiplying as it is independent events

 

2*2*3*3=36