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State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
To do:
We have to find whether the given statements are true or false.
Solution:
(a) The sum of three odd numbers is odd.
Example:
$3, 5$ and $7$ are odd numbers.
$3 + 5 + 7 = 15$
15 is an odd number.
The given statement is false.
(b) The sum of two odd numbers and one even number is even.
Example:
$3, 5$ are odd numbers and $6$ is an even number.
$3 + 5 + 6 = 14$
14 is an even number.
The given statement is true.
(c) The product of three odd numbers is odd.
Example:
$3, 5$ and $7$ are odd numbers.
$3 \times 5 \times 7 = 105$
105 is an odd number.
The given statement is true.
(d) If an even number is divided by 2, the quotient is always even.
Example:
$12$ is an even number.
$12 \div 2 = 6$
6 is an even number.
The given statement is false.
(e) All prime numbers except 2 are odd.
Example:
$2$ is an even prime number.
The given statement is false.
(f) Prime numbers are those numbers that are divisible by themselves and 1. No other number or numbers can divide them. These numbers have only two factors which include 1 as well.
The given statement is false.
(g) Sum of two prime numbers is not always even.
Example:
$2$ and $3$ are prime numbers.
$2+3=5$
$5$ is an odd number.
The given statement is false.
(h) Prime numbers are those numbers that are divisible by themselves and 1.
2 has only two factors(1 and 2).
This implies,
$2$ is a prime number.
Every other even number is divisible by 2 which means they have 2 as a factor.
The given statement is true.
(i) Composite numbers are those numbers that are NOT prime numbers. These numbers have three or more factors which include 1 as well.
Prime numbers are those numbers that are divisible by themselves and 1.
2 has only two factors(1 and 2).
This implies,
$2$ is an even prime number.
The given statement is false.
(j) The product of two even numbers is always even.
Example:
6 and 12 are even numbers
$6\times12=72$
72 is an even number
The given statement is true.
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