Number Properties – (Advanced) – Odd/Even Factorials

Today we will discuss how factorials affect the behavior of odd and even integers. Since we are going to deal with factorials, positive integers will be our concern. Using a question, we will see how factorials are divided.

Question: If  and  are positive integers, is  odd?

Statement 1:  = odd

Statement 2:  is greater than 2

Solution: The question stem doesn’t give us much information – just that  and  are positive integers

Question: Is  odd?

Statement 1:  = odd

Note that odd and even are identified only for integers. Since  is odd, it must be a positive integer. This means that ! must be equal to or less than 

Now think, how are  and  related? If  is odd,  is even and hence  is odd. If  is even, by the same logic,  is even.

Note that  and  have common factors starting from 1. Since  is less than or equal to ,  will be less than or equal to . So all factors in the denominator, from 1 to  will be there in the numerator too and will get cancelled leaving us with the last few factors of 

To explain this, let us take a few examples:

Example 1: Say, , 

Example 2: Say, , 

 (only one leftover factor)

Example 3: Say, , 

 (two leftover factors)

If the division of two factorials is an integer, the factorial in the numerator must be larger than or equal to the factorial in the denominator.

So what does  is odd imply? It means that the leftover factors must be all odd. But the leftover factors will be consecutive integers. So after one odd factor, there will be an even factor. If we want  to be odd, we must have either no leftover factors (such that ) or only one leftover factor and that too odd.

If we have no leftover factor, it doesn’t matter what  is as long as it is equal to . It could be odd or even. If there is one leftover factor, then  must be odd and hence  must be odd. Hence  could be odd or even. This statement alone is not sufficient.

Statement 2:  is greater than 2

This tells us that  is not equal to  since  is not 1. But all we know is that it is greater than 2. It could be anything as seen in examples 2 and 3 above. This statement alone is not sufficient.

Both statements together tell us that  is greater than  such that  is odd. So there must be only one leftover factor and it must be odd. The leftover factor will be the last factor i.e. . This tells us that  must be odd. Hence y must be odd too.

Answer (C).

Takeaways: Assuming  and  are positive integers,

–  will be an integer only if 

–  will be an odd integer whenever  or  is odd and 

–  will be an even integer whenever  is even and 

Think about this: what happens when we put 0 in the mix?