Counting Factors

Is there an efficient way to work out how many factors a large number has?

Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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

This printable worksheet may be useful: Power Mad

"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.