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Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3. - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 2

Question 2

Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3.

Worked Solution & Example Answer:Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3. - Edexcel - A-Level Maths Pure - Question 2 - 2020 - Paper 2

Step 1

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Answer

Let n be expressed as n = 3k, where k is a natural number. Then, the square of n is calculated as:

$n^2 = (3k)^2 = 9k^2$

Since 9k^2 is clearly a multiple of 3, this fulfills the condition.

Step 2

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Answer

Now, let n be expressed as n = 3k + 1. The square of n is:

$n^2 = (3k + 1)^2 = 9k^2 + 6k + 1$

This can be rewritten as:

$n^2 = 3(3k^2 + 2k) + 1$

This shows that n^2 is one more than a multiple of 3.

Step 3

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Answer

Lastly, let n be expressed as n = 3k + 2. The square of n is:

$n^2 = (3k + 2)^2 = 9k^2 + 12k + 4$

This can be rewritten as:

$n^2 = 3(3k^2 + 4k + 1) + 1$

This also indicates that n^2 is one more than a multiple of 3.

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