### Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### 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.

# Back to Basics

##### Age 14 to 16 Challenge Level:

In March we posed the problem:
"The number $3723$ (in base $10$) is written as $123$ in another base.
What is that base?" ............ The answer to this can be found in the March problem archive.

We could have written this question as:

Find b where $3723_{10} = 123_{b}$

So, moving on ...................

$123_{20}$ is $1 \times 20^2 + 2 \times 20 + 3 = 443_{10}$

$123_{21}$ is $1 \times 21^2 + 2 \times 21 + 3 = 486_{10}$

$123_{22}$ is $1 \times 22^2 + 2 \times 22 + 3 = 531_{10}$

$531 - 486 = 45$

$486 - 443 = 43$

Investigate these differences when $123_{b}$ is converted to base $10$ (for different values of $b$).

Try to explain what is happening.