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a) Show that n(2n + 1)(7n + 1) is divisible by 6 for all integers n.

b) Find all integers n such that n(2n + 1)(7n + 1) is divisible by 12.

 Sep 1, 2020
 #1
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You can use modular arithmetic.

 Sep 1, 2020
 #2
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Please put only one question per post.

 

a)  I would prove this using induction.     I will outline the method here.

 

Step 1:

Show true for n=0

 

Step 2:

Assume true for n=k where k is some integer.

then prove that it will be true for n=k+1   (I've done this myself so I know it works)

 

Step 3

Assume true for n=k

Prove it is true for n=k-1

 

Step 4.  (Summary)

Since it is true for n=0 it must be true for n=1, n=2 .......etc

also, since it is true for n=0 it must be true for n=-1, n=-2 ..... etc

Therefore it must be true for all integer values of n.

 

 

Now try filling in the blanks.

 Sep 1, 2020

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