Read the following statements carefully and select the correct option.

Statement-I: A number for which sum of all its factors is equal to the twice of the number is called a perfect number.

Statement-II: If a number is divisible by another number, then it is divisible by each of the factors of that number.

A. Both Statement-I, and Statement-II are true.
B. Statement-I is true, but Statement-II is false.
C. Statement-I is false, but Statement-II is true.
D. Both Statement-I, and Statement-II are false.

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Solution

i) A perfect number is such a number for which the sum of all its factors is equal to the twice of the number.
Example: 6 is a perfect number.
Factors of 6 are 1, 2, 3 and 6.
Sum of all its factors = 1 + 2 + 3 + 6 = 12 = 2 $\times$ 6.
Thus, Statement-I is correct.

ii) If a number is divisible by another number, then it is divisible by each of the factors of that number. Let's check this statement through the following example.

Take a number 96 which is divisible by 48.
Now check whether 96 is divisible by all the factors of 48.
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
96 is divisible by each of the factors of 48.
Thus, the statement-II is also correct.

Hence, the correct option is A.

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Statement I

n \geq 2

\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} > \sqrt{n}

Statement II

n \geq 2

\sqrt{n (n + 1)} < n + 1

6 n^{2} + 3 n + 1

a n^{2} + b n

a

b

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