12/13/2017
OCR Further Maths Additional Pure Book
5992 + 60 = 6052 Repeat
If the new number is too large repeat the process.
605 + 2 × 12 = 629 62 + 9 × 12 = 170 = 17 × 10 Hence 17 | 59925
E
Tip
PL
It is a common error to use the value of k rather than the value of n when using the algorithm. If the algorithm fails, go back and check that you have used the correct value of n.
WORKED EXAMPLE 2.10
SA M
By putting n = 4q + r, where n is any odd integer, prove that the square of n takes the form 8k + 1, where k is an integer. By the division algorithm, let the integer n be given by n = 4q + r.
Since n is odd then r = 1 or r = 3.
As n is odd then r must be odd and 0 ≤ r < 4.
When r = 1 : n 2 = (4q + 1) 2
[Check with, say, n = 11 : 11 2 = 121 = 8 × 15 + 1]
= 16q 2 + 8q + 1
= 8(2q 2 + q) + 1 = 8k + 1
When r = 3 : n 2 = (4q + 3) 2 = 16q 2 + 24q + 9
= 8(2q 2 + 3q + 1) + 1 = 8k + 1
Hence, the square of any odd integer is of the form 8k + 1.
Euclid’s algorithm To find the highest common factor of a and b(a > b). file:///C:/Profiles/j.raja/Desktop/New%20folder%20(2)/New%20folder%20(2)/P1.html
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