Question

State true (T) of false (F):

$\left(i\right)$ The sum of primes cannot be a prime.

$\left(ii\right)$ The product of primes cannot be a prime.

$\left(iii\right)$ An even number is composite.

$\left(iv\right)$ Two consecutive numbers cannot be both primes.

$\left(v\right)$ Odd numbers cannot be composite.

$\left(vi\right)$ Odd numbers cannot be written as sum of primes.

$\left(vii\right)$ A number and its successor are always co-primes.

Answer 1

$\left(i\right)$ The sum of primes cannot be a prime.

Ex: $2+3=5$ which is a prime number.

So, the given statement is False.

$\left(ii\right)$ The product of primes cannot be a prime.

Since, in the product of two primes, those two primes also factors of the product.

So, The product of prime numbers is always a composite number.

Ex: $7\times 5=35$

Factors of $35$ are $1,5,7$ and $35$.

So it is a composite number.

So, the given statement is True.

$\left(iii\right)$ An even number is composite.

The even number $2$ is not a composite number.

So, the given statement is False.

$\left(iv\right)$ Two consecutive numbers cannot be both primes.

$2$ and $3$ are consecutive numbers and are also prime numbers.

So, the given statement is False.

$\left(v\right)$ Odd numbers cannot be composite.

Ex: $9$ is an odd number but it is a composite number as its factors are $1,3$ and $9$.

So, the given statement is False.

$\left(vi\right)$ Odd numbers cannot be written as sum of primes.

$9$ is an odd number

$9=7+2$ where $7$ and $2$ are prime numbers.

So, Odd numbers can be written as sum of primes also.

So, the given statement is False.

$\left(vii\right)$ A number and its successor are always co-primes.

Co-primes: The common factor of two numbers is $1$.

A number and its successor have only one common factor $1$.

Ex: lets take two consecutive numbers $9$ and $10$.

Factors of $9$ $=1,3,9$

Factors of $10$ $=1,2,5,10$

$\Rightarrow$ common factor of $9$ and $10$ is $1$.

So, the given statement is True.