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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.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}.

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.