Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

The positive integers m and n are both odd. Which of the following integers is also odd?  In this question we can use our method of eliminating answers. So if we look at a first, just reading from left to right, we note that n plus n will always be even  because n is an odd number. We also know that any even number times an odd number will be even. Therefore, a is even. In b, as we previously determined, n plus one was even.  Therefore, regardless if m plus one is even or odd, and it is even, then it will still be an even number. Therefore, b is also even.  For c, m is odd, n is odd, and we add two. We know that two odd numbers added together will always be even. Therefore, the m plus n term will be even  and we add two more, so c will also be even.  In e, as we previously determined, m plus n are two odd numbers added together. Therefore, m plus n itself will be even and e is even. That leaves only d.  We can show that m times n is odd because any two odd numbers multiplied with each other will be odd. Adding an even number to an odd number will still give an odd number.  So d as a whole will be odd and it is our correct answer.

Video Summary

To determine which expression is odd given that \( m \) and \( n \) are both odd integers, we evaluate each option:<br /><br />- \( a: n(n + 1) \) is even since \( n + 1 \) is even.<br />- \( b: (m + 1)(n + 1) \) is even since both terms are even.<br />- \( c: m + n + 2 \) is even because \( m + n \) is even, and adding 2 keeps it even.<br />- \( e: m + n \) is even because the sum of two odd numbers is even.<br /><br />This leaves \( d: mn + 1 \). Here, \( mn \) is odd (the product of odd numbers is odd), and adding 1 gives an odd number. Therefore, \( d \) is the correct odd expression.

Keywords

odd integers

expression evaluation

mathematics

odd and even numbers

mn plus one