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Divisors Friendly Numbers Sociable Numbers Quadratic Formula

Divisors Friendly Numbers Sociable Numbers Quadratic Formula. Divisors. Divisors are all the numbers which divide evenly into a number including the number 1, but not including the number itself in the case of friendly and sociable numbers Therefore the divisors of 24 are

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Divisors Friendly Numbers Sociable Numbers Quadratic Formula

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  1. Divisors Friendly Numbers Sociable Numbers Quadratic Formula

  2. Divisors • Divisors are all the numbers which divide evenly into a number including the number 1, but not including the number itself in the case of friendly and sociable numbers • Therefore the divisors of 24 are 1, 2, 3, 4, 6, 8 and 12 • They sum to 36

  3. Friendly Numbers • These are pairs of numbers such that each number is the sum of the divisors of the other number. • Talismen sold in the middle ages would be inscribed with these numbers, on the grounds that they would promote love. • An Arab mathematician claims that people would write one of the pair of numbers on one fruit and eat it, writing the second number on another fruit and give it to a lover as a mathematical aphrodisiac!

  4. Friendly Numbers • There was only one pair discovered until Fermat discovered the pair 17,296 and 18,416 in 1636 • Descartes discovered the pair 9,363,584 and 9,437,056 in 1638 • In 1866 a sixteen year old Italian found the pair 1184 and 1210 • Computers can be programmed to find larger ones now!

  5. Project • The divisors of 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 • Sum = 1210 • The divisors of 1210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 • Sum = 1184

  6. Sociable Numbers • These are groups of three or more numbers which form closed loops • The sum of the divisors of the first give the second • The sum of the divisors of the second give the third and so on until the divisors of the last give the first number

  7. –––––– –b±b – 4ac ––––––––––––– x = 2a Quadratic formula • The Quadratic formula works to factorise all equations of the form ax2+bx+c= 0 • The roots are:

  8. 2 b- 4ac -b± x = 2a -(9)± (9)2 -4(1)(20) x = -9 ± 81–80 2(1) = 2 -9 ± 1 = 2 -10 2 -8 2 -9 ± 1 or = = 2 x2+ 9x+ 20 = 0 a= 1 b= 9 c= 20 x=-4 or –5

  9. The area of a rectangle is 77 cm2. One side is 4 cm longer than the other. Find the length and breadth of the rectangle. x+ 4 x Area = 77 cm2 x(x + 4x) = 77 x2+ 4x– 77 = 0

  10. 2 b- 4ac -b± x = 2a -(4)± (4)2 -4(1)(–77) x = - 4 ± 324 - 4 ± 16+308 2(1) = = 2 2 14 2 -22 2 - 4±18 = or = 2 x2+ 4x– 77 = 0 a= 1 b= 4 c=–77 x=7 or –11

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