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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

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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}.

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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

2-digit Square

Age 14 to 16 Challenge Level:

Why do this problem?

This problem provides reinforcement of the concept of place value and experience of reading the words in a question and forming an algebraic expression using the information given. It also provides practice in algebra involving the difference of two squares, factorising and solving linear simultaneous equations.

Possible approach

If you think the class will not remember having learnt the difference of two squares the class could first work on and discuss Plus Minus. However this problem leads naturally into the difference of two squares without the learner having to recognise it at first so it could provide a useful reminder in itself. The learners could first work individually to give them 'thinking time', then work in pairs to support each other and to give an opportunity for mathematical talk, and finally there could be a class discussion.

Key questions

Give an example of a 2-digit number . [e.g. 27]

What place value does each digit hold/stand for? [2 tens, 7 units]

Fill in the blank: 27 = 2 times____+ 7

If the digits are reversed what will the new number be? [72]

If a 2 digit number has tens digit a and units digit b then the number is ___times a + ___?

If you know a number is a square what can you say about its factors?

Possible extension
What's Possible? is another non-standard problem involving the difference of two squares.

Possible support
The problem Plus Minus is a little easier.