Question

The product of two odd numbers is
(a) An even number
(b) An odd number
(c) Can’t be determined
(d) None of these

Answer 1

Hint: To find if the product of two odd numbers is an odd number or an even number, consider any two odd numbers of the form \[2n+1\] and \[2m+1\] where m and n are integers. Multiply them and simplify the expression to check if the product is an odd number or an even number.

Complete step-by-step answer:
We have to find if the product of two odd numbers is an even number or an odd number.
Let us consider any two odd numbers of the form \[2n+1\] and \[2m+1\] where m and n are integers.
We will now multiply the two numbers.
Thus, we have \[\left( 2n+1 \right)\left( 2m+1 \right)=4mn+2n+2m+1\].
By taking out the common terms, we can rewrite the above expression as \[\left( 2n+1 \right)\left( 2m+1 \right)=4mn+2n+2m+1=2\left( 2mn+n+m \right)+1\].
Let us assume \[k=mn+m+n\]. We observe that ‘k’ is an integer as well because the sum and product of any two integers is also an integer.
Thus, we have \[\left( 2n+1 \right)\left( 2m+1 \right)=2\left( 2mn+n+m \right)+1=2k+1\]. Here, \[2k+1\] represents another odd number.
Hence, the product of any two odd numbers is also an odd number, which is option (b).

Note: We can also check that the product of two odd numbers is odd by considering any two odd numbers and multiplying them to see if their product is an odd number or an even number. Odd numbers are those numbers which can’t be divided exactly into pairs, i.e., they leave a remainder when they are divided by 2. Odd numbers have the digits 1, 3, 5, 7 and 9 at their units place. Even numbers are those numbers which are exactly divisible by 2. Even numbers can have the digits 0, 2, 4, 6 and 8 at their units place.