12/13/2017
20
OCR Further Maths Additional Pure Book
If p is an odd prime number and q and p have no common factors, then Euler proved that q is a quadratic residue of p if and only if q ( p − 1 ) / 2 ≡ ± 1(modp). Demonstrate that this result is correct for all the quadratic residues of modulo 11. You will be able to prove this result when you have studied the next section.
E
Section 4: Prime numbers Key point 2.4
Prime: an integer p (p ≠ 1) is prime if it has no divisors except 1 and itself.
PL
Composite: a composite number has at least one divisor other than 1 and itself.
Highest common factor: the hcf (also called the greatest common divisor gcd) is the highest factor of two or more numbers. Coprime: two or more integers are coprime (or relatively prime) if 1 is there only common factor.
SA M
Fundamental theorem of arithmetic (also known as the unique prime factorisation theorem) states that every integer greater than 1 is either prime or the product of primes in exactly one way (apart from arrangements). Useful results for integers a, b and c:
If a and b are coprime and a | c and b | c, then ab | c. If a | b and c | d, then ac | bd. If a | b and b | c, then a | c. If a | b and a | c, then a | (bx + cy), where x and y are integers.
Bezout’s identity: the highest common factor of b and c is bx + cy. Note: if this hcf is 1 , then b and c are coprime.
WORKED EXAMPLE 2.20
Use Bezout’s identity to show that 8 is the highest common factor of 40 and 56.
8 = 3 × 40 − 2 × 56 Hence 8 is the hcf(40, 56)
List multiples of each number: 40, 80, 120, 160, … 56, 112, 168, 224, … Spot the linear combination.
Euclid’s algorithm to find the hcf(56, 40) : file:///C:/Profiles/j.raja/Desktop/New%20folder%20(2)/New%20folder%20(2)/P1.html
Original material © Cambridge University Press 2018
63/130