# Thread: Proof by induction that 4| 5^n - 1

1. ## Proof by induction that 4| 5^n - 1

I have this problem that I know ( by just knowing ) that its true, but I have to prove by induction, and I am stuck on the induction step.

Use induction to show that 5^n - 1 (5 to the power of n ) is divisible by 4 for all n element of Natural numbers , n >= 1.

So I have come this far:

Basic Step:

Let P(n) be the predicate 4 | 5^n - 1

Then P(1) is 4|5^1 - 1 = 4 which is a tautology

Induction Step:

We must show that for k>=1 if P(k) is true, then P(k+1) must also be true.
We assume for some k >=1 that

4| 5^n - 1

We now want to show that P(k+1) is true:

4 |5^(k+1) - 1

The right hand side of this statement is

5^(k+1) - 1 = 5^k.5 - 1

Now I know that it will work for all exponents of 5 because every exponential of 5 ends in 25 ( 125, 625, 3125, 15625) and minusing 1 gives 24 which is divisble by 4 ( and any multiple of 100 is also divisble by 4)
So how can i put this in induction language?

I have also tried it this way:

Let's say
5k - 1 = 4r where r is a natural number. We can rewrite this as :
5^k = 4r + 1

Basic step:
So for n=1 P(1) = 5-1=4(1) = 4,this is true.

Induction step:
Then for k+1:

5k+1 - 1 = 4r

5.5k = 4r + 1

This is the same as above (in bold) showing that for k+1 5k-1 is always divisible by 4.

Many thanks

2. $\displaystyle 5^{k+1}-1=5^{k+1}-5+5-1=5(5^k-1)+(4)$

3. Originally Posted by iva
I have this problem that I know ( by just knowing ) that its true, but I have to prove by induction, and I am stuck on the induction step.

Use induction to show that 5^n - 1 (5 to the power of n ) is divisible by 4 for all n element of Natural numbers , n >= 1.

So I have come this far:

Basic Step:

Let P(n) be the predicate 4 | 5^n - 1

Then P(1) is 4|5^1 - 1 = 4 which is a tautology

Induction Step:

We must show that for k>=1 if P(k) is true, then P(k+1) must also be true.
We assume for some k >=1 that

4| 5^n - 1

We now want to show that P(k+1) is true:

4 |5^(k+1) - 1

The right hand side of this statement is

5^(k+1) - 1 = 5^k.5 - 1 $\displaystyle \color{red}=(4)5^k+5^k-1$

Now I know that it will work for all exponents of 5 because every exponential of 5 ends in 25 ( 125, 625, 3125, 15625) and minusing 1 gives 24 which is divisble by 4 ( and any multiple of 100 is also divisble by 4)
So how can i put this in induction language?

Many thanks
If $\displaystyle 5^k-1$ is divisible by 4, then $\displaystyle (4)5^k + \left(5^k-1\right)$ will definately be divisible by 4.

4. Hello, iva!

Use induction to show that: .$\displaystyle 5^n - 1$ is divisible by 4 . for all $\displaystyle n \in N,\;n \geq 1$

You proved the basic statement: .$\displaystyle S(1)$ is true.

We assume that $\displaystyle S(k)$ is true:

. . $\displaystyle 5^k - 1 \:=\:4a\;\text{ for some integer }a$

Add $\displaystyle 4\cdot5^k$ to both sides:

. . .$\displaystyle 5^k + {\color{blue}4\cdot5^k} -1 \;=\;4a + {\color{blue}4\cdot5^k}$

. . $\displaystyle (1+4)\cdot5^k -1\;=\;4a + 4\cdot5^k$

. . . . . $\displaystyle 5\cdot5^k - 1 \;=\; 4a + 4\cdot5^k$

And we have:

. . $\displaystyle 5^{k+1} -1\;=\;\underbrace{4(a + 5^k)}_{\text{multiple of 4}}$

We have proved $\displaystyle S(k+1)$
. . The inductive proof is complete.

5. Thanks a mill guys

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# prove that 5^n-1 is divisible by 4

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