State whether the following statements are True or False

Question – 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.

Solution –

Even numbers: Even numbers are divisible by 2 without leaving any remainder.

Odd numbers: Odd numbers are divisible by 2 with leaving a remainder.

Prime numbers : Prime numbers are the numbers with exactly two factors i.e., 1 and itself.

Composite numbers : Composite numbers  has more than two factors

(a) The sum of three odd numbers is even.

False ⇒ The sum of two odd numbers gives an even number. Further, the sum of the even number and the third odd number is an odd number.

For example : 5 + 7 + 3 = 15

Here, 5, 7 and 3 are odd numbers and their sum 15 is also an odd number.

Hence, the sum of three odd numbers is odd.

(b) The sum of two odd numbers and one even number is even.

True ⇒ The sum of two odd numbers is even. Further the sum of this even number with one even number is also even.

For example : 7 + 9 + 2 = 18

Here, 7 and 9 are odd numbers but their sum i.e., 16 is even.

Further, 2 and 16  even numbers. Hence, their sum is an even number.

(c) The product of three odd numbers is odd.

True ⇒ The product of two odd numbers is odd. Further the product of this odd number with another odd number is also odd.

For example : 3 × 5 × 7 = 105

Here, 3, 5 and 7 are odd numbers and their product 205 is also an even number.

(d) If an even number is divided by 2, the quotient is always odd.

False ⇒ Any even number when divided by 2, gives an even or odd quotient not always an odd quotient.

For example : 16 ÷ 2 = 8

Here, 16 is an even number and when divided by 2 gives quotient 8 which is an even number.

(e) All prime numbers are odd.

False ⇒ All prime numbers are not odd.

For example: 2 is an even prime number.

(f) Prime numbers do not have any factors.

False ⇒ Prime numbers have two factors, 1 and the number itself.

For example: 23 is a prime number and its factors are 1 and 23.

(g) Sum of two prime numbers is always even.

False ⇒ Since, 2 is a prime number. Its addition to any other odd prime number gives an odd number.

For example: 2 + 3 = 5

Here, 2 and 3 are prime numbers and their sum 5 is an odd number.

(h) 2 is the only even prime number.

True ⇒ 2 is the only even prime number.

(i) All even numbers are composite numbers.

False ⇒ 2 is an even prime number.

(j) The product of two even numbers is always even.

True ⇒ The product of two even numbers is always even.

For example: 2 × 4 = 8

Here, 2 and 4 are even numbers and their product 8 is also an even number.

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