# Is $N_{10}$ even or odd?

• MHB
• Ackbach
In summary, there is a simple rule for determining whether a number is even or odd - an even number is divisible by 2, while an odd number is not. $N_{10}$ can only be either even or odd, not both. There is a pattern for even and odd numbers, with even numbers always having a factor of 2 and odd numbers having a remainder when divided by 2. Knowing whether $N_{10}$ is even or odd can be useful in mathematical and scientific applications. Odd numbers can be negative, as long as they are not divisible by 2.
Ackbach
Gold Member
MHB
Here is this week's POTW:

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Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots +1/a_n=1$. Determine whether $N_{10}$ is even or odd.

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Re: Problem Of The Week # 230 - Aug 25, 2016

This was Problem A-5 in the 1997 William Lowell Putnam Mathematical Competition.

No one answered this week's POTW. The solution, attributed to Kiran Kedlaya and his associates, follows:

We may discard any solutions for which $a_1 \neq a_2$, since those come in pairs; so assume $a_1 = a_2$. Similarly, we may assume that $a_3 = a_4$, $a_5 = a_6$, $a_7 = a_8$, $a_9=a_{10}$. Thus we get the equation
$2/a_1 + 2/a_3 + 2/a_5 + 2/a_7 + 2/a_9 = 1.$
Again, we may assume $a_1 = a_3$ and $a_5 = a_7$, so we get $4/a_1 + 4/a_5 + 2/a_9 = 1$; and $a_1 = a_5$, so $8/a_1 + 2/a_9 = 1$. This implies that $(a_1-8)(a_9-2) = 16$, which by counting has 5 solutions. Thus $N_{10}$ is odd.

## 1. Is there a specific method for determining whether $N_{10}$ is even or odd?

Yes, there is a simple rule for determining whether a number is even or odd. An even number is divisible by 2, while an odd number is not. So, to determine if $N_{10}$ is even or odd, we just need to divide it by 2 and check if there is a remainder or not.

## 2. Can $N_{10}$ be both even and odd?

No, $N_{10}$ can only be either even or odd, it cannot be both. This is because even and odd numbers are two different categories, with even numbers having a specific set of characteristics and odd numbers having a different set of characteristics.

## 3. Is there a pattern for even and odd numbers?

Yes, there is a pattern for even and odd numbers. Even numbers always end in 0, 2, 4, 6, or 8, while odd numbers always end in 1, 3, 5, 7, or 9. This pattern continues for all numbers, with even numbers always having a factor of 2 and odd numbers having a remainder when divided by 2.

## 4. What is the significance of knowing whether $N_{10}$ is even or odd?

Knowing whether $N_{10}$ is even or odd can be useful in many mathematical and scientific applications. For example, in computer programming, determining whether a number is even or odd can help with creating efficient algorithms and in physics, even and odd numbers can have different properties when it comes to electrical charges and energy levels.

## 5. Can odd numbers be negative?

Yes, odd numbers can be negative. The definition of an odd number is that it is not divisible by 2, so it can be any integer that is not a multiple of 2, including negative numbers. For example, -5, -11, and -37 are all odd numbers.

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