Example 2

Find $v_5(11^{20}-1)$.

We apply LTE with

Since

and

LTE gives

Now

and

Therefore,

So $25$ divides $11^{20}-1$, but $125$ does not.

Example 3

Find $v_7(15^{14}-1)$.

Use LTE with

Since

we have

Also $7\nmid 15$, so the conditions hold.

As such

Now both $15-1$ and $14$ contain exactly one factor of $7$, so

and

Therefore,

So the largest power of $7$ dividing $15^{14}-1$ is $7^2$.

Sum Form

There is also a useful sum version of LTE.

Let $p$ be an odd prime. Suppose that

and

If $n$ is odd, then

This follows by applying the difference form to

When $n$ is odd,

so

Worked Examples for the Sum Form

Example 4

Find $v_3(8^9+1)$.

We use the sum form with

Since

and $n$ is odd, LTE gives

Now both $8+1$ and $9$ contain exactly two factors of $3$, so

and

Therefore,

So the largest power of $3$ dividing $8^9+1$ is $3^4$.

Example 5

Find $v_5(4^{15}+1)$.

Since

and $15$ is odd, we may apply the sum form:

Now

and

Thus

So $25$ divides $4^{15}+1$, but $125$ does not.

The Case $p=2$

The prime $2$ requires special care. The simplest useful version is the following.

If $a$ and $b$ are odd positive integers and $n$ is a positive integer, then

If $n$ is odd

If $n$ is even

This version is very useful, but it is also easier to misuse. Always check that $a$ and $b$ are odd before applying it.

Worked Examples for $p=2$

Example 6

Find $v_2(5^{12}-1)$.

Here

Since $a$ and $b$ are odd and $n$ is even, we use the even version:

Now

and

Therefore,

So the largest power of $2$ dividing $5^{12}-1$ is $2^4$.

Example 7

Find $v_2(7^9-1)$.

Here

Since $a$ and $b$ are odd and $n$ is odd,

Thus

So the largest power of $2$ dividing $7^9-1$ is $2$.

Using LTE in Olympiad Problems

LTE is most useful when a problem asks for:

• The highest power of a prime dividing an expression
• Divisibility by a large prime power
• Solving equations involving expressions like $a^n-b^n$
• Comparing valuations on both sides of an equation
• Proving that an expression is not a perfect power

A common pattern is to apply $v_p$ to both sides of an equation.

For example, if a problem gives

then applying LTE can turn the equation into a much simpler statement about

Key Contest Insight

When you see an expression like

or

and a prime divides $a-b$ or $a+b$, consider using LTE.

The key question is not just whether the prime divides the expression, but how many times it divides the expression.

LTE converts a difficult divisibility question into a simple valuation formula.

Strategy Tips

• Check whether the expression is of the form $a^n-b^n$ or $a^n+b^n$.
• Identify the prime $p$ whose power you want to count.
• Check whether $p\mid a-b$ for the difference form.
• Check whether $p\mid a+b$ and $n$ is odd for the sum form.
• Make sure $p\nmid ab$.
• Treat $p=2$ separately.
• After applying LTE, compute each valuation carefully.
• In equation problems, take $v_p$ of both sides.

Common Pitfalls

• Applying the odd prime version when $p=2$.
• Forgetting to check that $p\nmid ab$.
• Using the sum form when $n$ is even.
• Confusing $v_p(a-b)$ with $a-b$ itself.
• Forgetting that $v_p(n)$ contributes extra powers of $p$.
• Applying LTE before checking that $p$ divides $a-b$ or $a+b$.

Practice Problems