Since 32 + 42 = 52 does it follow that 33 + 43 + 53 = 63 ? Check it out ! Does this pattern continue to be true? The factors of 28 (not including itself) are 1, 2, 4, 7 and 14. Astonishingly these factors add up to give 28. Because 28 has this special property it is called a Perfect Number. 28 = 13 + 33 and so is also the sum of the first two odd cubes. Calculate 13 + 33 + 53 + 73. Find all the factors of this number. Is the number Perfect? Does this pattern continue to be true? Investigate. --- 
What have these odd triangles to do with squares and cubes?
Why are the numbers 1, 3, 6, 10, 15, 21 ... called Triangular Numbers? Try to extend this remarkable pattern: 3(1+3)3 = 13+33+2(14+34) = 192 3(1+3+6)3 = 13+33+63+2(14+34+64) = 3000 3(1+3+6+10)3 = ? = ? --- Continue with these calculations and .... be amazed! 1x9 + 2 = 11 12x9 + 3 = 111 123x9 + 4 = 1111 --- 6 has factors 1, 2, 3 and 6. How many factors does each these factors have? In order: 1, 2, 2 and 4. (1+2+2+4)2 = 13+23+23+43 30 has factors 1, 2, 3, 5, 6, 10, 15 and 30. How many factors does each of these factors have? In order: 1, 2, 2, 2, 4, 4, 4 and 8. ...and yes you've guessed it... (1+2+2+2+4+4+4+8)2 = 13+23+23+23+43+43+43+83 Are 6 and 30 the only such numbers....?? --- 1 + 5 + 10 + 18 + 23 + 27 = 2 + 3 + 13 + 15 + 25 + 26 12 + 52 + 102 + 182 + 232 + 272 = 22 + 32 + 132 + 152 + 252 + 262 13 + 53 + 103 + 183 + 233 + 273 = 23 + 33 + 133 + 153 + 253 + 263 Check that these astonishing equations are true. Do they hold for 4th powers and 5th powers and beyond? Be amazed by doing the calculations. --- 3 has factors 1 and 3 giving a total of 4 = 22 a square number: . 1501 = 19 x 79 a product of two primes and therefore has factors 1, 19, 79 and 1501. The total of these factors is 1600 = 402 a square number. Check that 22 also has this property. Can you find the other three numbers less than 100 that have this property? --- Let's multiply the first four positive integers together: 1x2x3x4 = 24. Obviously 5 does not divide 24 but it does divide 1 more than 24. Now let's try the first five positive integers multiplied together: 1x2x3x4x5 = 120. This time 6 does divide 120 and does not divide 1 more than 120. The next case is: 1x2x3x4x5x6 = 720. Does 7 divide this? Does 7 divide 1 more than 720?
Continue this table and observe something amazing about the pattern of yes/no answers in the last column.n 1x2x...x(n-1) divides by n?
1x2x...x(n-1) + 1 divides by n?
2 1 no 1+ 1 yes 3 1x2 no 1x2 + 1 yes 4 1x2x3 no 1x2x3 + 1 no 5 1x2x3x4 no 1x2x3x4 + 1 yes 6 1x2x3x4x5 yes 1x2x3x4x5 + 1 no 7 1x2x3x4x5x6 no 1x2x3x4x5x6 + 1 yes What has this to do with a man called Wilson? --- 3 x 37 = 111 and 1+1+1 = 3 6 x 37 = 222 and 2+2+2 = 6 9 x 37 = 333 and 3+3+3 = 9 ... continue this pattern and be amazed! ---
What property do the numbers 3 and 1501 share?
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