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If k is any even positive integer, then ${k^2} + 2k$ is
A) Divisible by 24
B) Divisible by 8 but may not be divisible by 24
C) Divisible by 4 but may not be divisible by 8
D) Divisible by 2 but many not be divisible by 4

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: In this question, we are given an unknown quantity k and we are also given an algebraic expression in terms of this unknown quantity. Using the properties of k, we have to determine which of the options satisfies the given expression. We are given that k is an even positive integer. So, using this information, we will write the given expression and find out the correct answer.

Complete step-by-step solution:
We are given that k is an even positive integer, so k is a multiple of 2.
So $k = 2n$ where n is a positive integer.
Putting this value in the given equation, we get –
$
  {k^2} + 2k = {(2n)^2} + 2(2n) \\
   \Rightarrow {k^2} + 2k = 4{n^2} + 4n \\
   \Rightarrow {k^2} + 2k = 4({n^2} + n) \\
 $
We see that ${k^2} + 2k$ is a multiple of 4 so it is divisible by 4, but we cannot say anything about its divisibility by 8 and 24.

Hence, option (C) is the correct answer.

Note: Real numbers are those numbers that can be shown on a number line, integers are a type of real numbers, these are defined as the numbers that are not in fractional or decimal form, for example, -4,0, 5,etc. are integers. There are two types of numbers, positive numbers and negative numbers, the numbers that are on the left side of the zero on the number line are called negative numbers and the numbers that are on the right side of the zero are known as positive numbers. And even numbers are those that are divisible by 2 or we can also say that even numbers are those numbers that are a multiple of 2.