Powerful factors
Use the fact that: x²-y² = (x-y)(x+y) and x³+y³
= (x+y) (x²-xy+y²) to find the highest power of 2 and the
highest power of 3 which divide 5^{36}-1.
Problem
Use the following identities:
$x^2-y^2 \equiv (x-y)(x+y)$and
$x^3+y^3 \equiv (x+y)(x^2-xy+y^2)$
to find the highest power of $2$ and the highest power of $3$ which divide $5^{36}-1$.
Getting Started
Factorise $5^{36}-1$ into as many factors as you can, until you can calculate the values and see which ones are even and which are multiples of $3$.
Student Solutions
The problem required the use of the facts that
to find the highest power of 2 and the highest power of 3 which divide $5^{36}-1.$
Alexander Marynovsky from Israel sent in this solution.
Remember what we have got here, I'll use it twice (for 2 and for 3).
Now let's take out the 2's from it.
Because $5^n - 5^k$ is even and therefore $5^n - 5^k +1$ is odd, $(5^2- 5+1), (5^6- 5^3 +1), (5^4 - 5^2 +1), (5^{12} -5^6 +1)$ obviously can't be divided by 2.
So we are left with
So the highest power of 2 is 4.
Combining these results:
The highest power of 3 is 3.
The method can be shortened using modulus arithmetic.
Teachers' Resources
Why do this problem?
For practice in factorising polynomials.Key question
What is the highest power of 5 we can find using a calculator?Can we factorise this expression to get factors involving smaller powers of 5, so that all the powers of 5 can be found using a calculator?
Howdo you know if a number is divisible by 3?