Question

Read the following statements.
Statement-1 : A number for which sum of all its factors is equal to twice the number is called perfect number.
Statement-2 : If two numbers are divisible by a number, then their sum and difference are also divisible by that number.
Which of the following options holds?

• A. Both Statement-1 and Statement-2 are true.
• B. Statement-1 is true but Statement-2 is false.
• C. Statement-1 is false but Statement-2 is true.
• D. Both Statement-1 and Statement-2 are false.

Answer 1

The correct option is A Both Statement-1 and Statement-2 are true.

Statement $1$ is true.

As by definitions, perfect number is positive integers which is equals to the sum of its proper positive divisors.

$\therefore$ (sum of proper divisor)$+$(self)$=$ sum of all divisor.

as [proper divisor's sum$=$ self number] in case of perfect number

$\therefore$ self $+$ self $=$ sum of all divisors

$\therefore$ sum of all divisors $=$ 2 x perfect number.

Statement $2$ is true.

Let the two number $x$ and $y$ is divisible by $k$ having quotient $a$ and $b$

$\therefore$ $x=ak$

$y=bk$

sum$=ak+bk=(a+b)k$ is divisible by $k$.

Difference $=ak-bk(a-b)k$ is divisible by $k$.

The correct option is A both the statement $1$ and statement $2$ are true.