### N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

### Why do this problem?

This problem offers practice in working with indices to develop fluency, while providing an intriguing context to discover patterns and find justifications.

### Possible approach

"Work out and write down the powers of $2$ from $2^1$ up to $2^8$." Give students a short time to do this, perhaps using mini-whiteboards.
"What do you think would be the last digit of $2^{100}$?" Give students time to discuss this with their partner before sharing ideas and justifications.
"Are there any powers of two that are multiples of $10$?" "No, because a power of 2 has to end in a 2, 4, 6 or 8, and a multiple of 10 ends in a 0".

For the next part of the lesson, you could divide the class into pairs or small groups, and give each group one of the following to work on:
• Work out $2^n+3^n$ for some different odd values of $n$.
What do you notice?
What do you notice when $n$ is a multiple of $4?$

• For which values of $n$ is $1^n + 2^n + 3^n$ even?

• Work out $1^n + 2^n + 3^n + 4^n$ for some different values of $n$.
What do you notice?

• Work out $1^n + 2^n + 3^n + 4^n + 5^n$ for some different values of $n$. What do you notice?
When students have finished working on their question and justified their findings, invite them to look for similar results of their own. Here are a few suggestions that they could explore for different values of $n$:
$4^n + 5^n + 6^n$
$3^n + 8^n$
$2^n + 4^n + 6^n$
$3^n + 5^n + 7^n$
$3^n - 2^n$
$7^n + 5^n - 3^n$

To finish off, students could present their findings to the rest of the class, with emphasis on clear explanations to justify that the patterns they have found will continue for all values of $n$.

### Key questions

What patterns can you find in the units digit of ascending powers of 2, 3, 4...?
How can you be sure the patterns will continue?

### Possible extension

This is an open ended activity which already offers plenty of opportunities for extension work.
The Stage 5 problem Tens takes the ideas in this problem and treats them in a more formal way, encouraging the use of Modular Arithmetic notation.

### Possible support

You might suggest that students draw up 'power tables' so that the cyclical nature of the units digits becomes apparent.