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## 'Double Time' printed from http://nrich.maths.org/

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To encode the following message the text was first written as pairs of letters ignoring the spaces between the words. For example John Brown would be written jo hn br ow nz where the z is added because there are an odd number of letters.

The letters of the alphabet are numbered $a=0,\ b=1,\ {\rm to}\ z=25$ and each pair of letters is replaced by a pair of numbers. There are $26\times 26=676$ pairs of numbers. It is possible to store the codes for all 676 pairs and look them up as needed but much more efficient to use a decyphering formula.

Each pair of numbers $(\alpha, \beta)$ is encoded as another pair of numbers $(\alpha',\beta')$ where $$\alpha' = \alpha + 3\beta \pmod {26}$$ $$\beta' = 5\beta \ \ \pmod{26}$$ Find $\alpha$ and $\beta$ in terms of $\alpha'$ and $\beta'$ and hence decode the following quotation which is a remark made by Einstein:

dj lb rn qm bu ao hd eo kr ia cs ud rx cm qo bn fr ld ek th ys wm