1714 views

### Prove by induction that 11^n - 6 is divisible by 5 for all positive integer n.

Let P(n) be the statement that 11n - 6 is divisible by 5.

BASE CASE: Let n = 1.  This gives 111 - 6 = 5, obviously divisible by 5, therefore we know that P(1) is true.

HYPOTHOSIS STEP: Assume that P(k) is true for some positive integer k.  We can write this a different way: 11k – 6 = 5m where m is also a positive integer.

INDUCTION STEP: We will now show that P(k+1) is true.

P(k+1) states that 11k+1 – 6 is divisible by 5.

11k+1 – 6 = 11 * (11k) – 6

= 11* (6 + 5m) – 6  (now we use our hypothesis step, with rearranged expression 11k = 6 + 5m)

= 55m + 60   (multiplying out the brackets gives)

= 5 * (11m + 12)  (now factorising again)

Which shows that this is a factor of 5 and that P(k+1) is true.

CONCLUSION: Since P(k+1) is true given P(k), and we know that P(1) is true, we have proved by induction that P(n) is true for all positive integer n.

1 year ago

Answered by Sally, who has applied to tutor A Level Further Mathematics with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 149 SUBJECT SPECIALISTS

£26 /hr

Degree: Mathematics MMath (Masters) - Manchester University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“Hello there, I'm Mark and I study Mathematics at the University of Manchester! Let me help you achieve your best!”

£30 /hr

Degree: General Engineering (Masters) - Durham University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“I study General Engineering at the University of Durham. I have always enjoyed maths and sciences, so hope to help students who share my love for them too!”

£24 /hr

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“I’ll help anyone finding maths hard. I still struggle now, it's not meant to be easy! Be at my tutorial or (c^2-a^2) <- a maths joke, I apologise.”

£20 /hr

Degree: Physics (Bachelors) - Bristol University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“About Me Hi, I'm Sally and I'm currently studying physics at the University of Bristol.  Degree level physics has given me not only an in-depth knowledge in the science but also strong skills in using maths not just for its own sake b...”

MyTutor guarantee

### You may also like...

#### Posts by Sally

Find the magnitude of the force on an electron that is travelling with velocity 2 x 10^4 ms^(-1) in the x direction through a uniform magnetic field of strength 2T in the y direction.

If I have £730 in my bank account which has 2.5% compound interest per year, how much more money will be in there after two years?

Prove by induction that 11^n - 6 is divisible by 5 for all positive integer n.

#### Other A Level Further Mathematics questions

Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).