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n is a positive integer - OCR - GCSE Maths - Question 15 - 2018 - Paper 1

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Question 15

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n is a positive integer. Prove that $13n + 3 + (3n - 5)(2n + 3)$ is a multiple of 6.

Worked Solution & Example Answer:n is a positive integer - OCR - GCSE Maths - Question 15 - 2018 - Paper 1

Step 1

Calculate $13n + 3 + (3n - 5)(2n + 3)$

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Answer

First, expand the expression:

  1. Calculate (3n−5)(2n+3)(3n - 5)(2n + 3):

    (3n−5)(2n+3)=6n2+9n−10n−15=6n2−n−15(3n - 5)(2n + 3) = 6n^2 + 9n - 10n - 15 = 6n^2 - n - 15

  2. Now substitute back into the original expression:

    13n+3+(6n2−n−15)13n + 3 + (6n^2 - n - 15)

    Combine the like terms:

    6n2+(13n−n)+(3−15)=6n2+12n−126n^2 + (13n - n) + (3 - 15) = 6n^2 + 12n - 12

Step 2

Show that the expression is a multiple of 6

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Answer

The expression simplifies to:

6n2+12n−126n^2 + 12n - 12

We can factor out a 6:

6(n2+2n−2)6(n^2 + 2n - 2)

Since the expression is a product of 6 and another integer (n2+2n−2)(n^2 + 2n - 2), it proves that the overall expression is indeed a multiple of 6.

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