By A. Gardiner

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31 14. Find a prime number of the form 3" - 15. Find another prime number of the form 3" - 1. 16. Find a prime number of the form 3" + 1. 1. 1. 17. Find a prime number of the form 4" 18. Find another prime number of the form 4" - 19. What can you say about n when 4" 1. + 1 is a prime number? 20. (a) What can you say about m and n when mil number? (b) What can you say about m and n when mil number? is a prime + is a prime Investigation (a) Find the first prime number which is exactly 9 less than a perfect square.

Ii) Once you discover which digit is wrong, rub it out. One of the two corrco:t digits should tell you the tens digit of the number you are dividing by. The rest requires a little more thought, but is not too hard. (b) If you managed the one in part (a), then you should not have too much trouble with the first two here. ) 5. (a) Look at the first step: 'bed - gh = ;'. What does this tell you about b, e, and g? Then's x a7 = gh' leaves just two possibilities for a. You have to decide which of these two possibilities fits the other facts.

There are in fact five different ways of marking a 12 unit ruler with this smallest conceivable number of marks so that each length from 1 unit up to 12 units can be measured directly. ) 44 Extension Suppose you have a 12 by 12 sheet of transparent plastic. If you draw a grid of eleven equally spaced vertical lines and eleven equally spaced horizontal lines you would get an 'area measurer' which could measure directly any x by y rectangle, where x and yare any whole numbers up to 12. There would then be 144 ways of measuring a 1 by 1 square directly.