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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Adding All Nine

### Why do this
problem?

### Key questions

### Possible extension

More able learners could explore what multiples of $9$ they can and
cannot make using all the digits $1$ to $9$ once and once only.
These will be between $45$ (the result of adding all nine digits as
$1$-figure numbers) and $987654321 + 1$. Repeat with he set of
numbers $1$ to $8$.

### Possible support

Suggest finding different $2$-digit numbers the set of digits $1$
to $9$, and then total these adding in the 'extra' digit and work
from this total.

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem
requires plenty of accurate adding! Although the ability to do
division is called for, calculators could be used to perform the
operation as well as to check results.

The investigation leads learners to generate for themselves
the rule for divisibility by $9$ - that if the digits in a number
add to $9$ or a multiple of $9$.

Have you checked your adding?

Is this number a multiple of $9$?

Have you checked using a calculator?

How many $2$-digit numbers have you found that are divisible
by $9$?

What happens if you just use the numbers from $1$ to
$8$?