12/13/2017
OCR Further Maths Additional Pure Book
d x ≡ 9(mod 15), x ≡ 10(mod 17), x ≡ 11(mod 20) 14
Solve the following simultaneous linear congruences: a
x ≡ 9(mod 12), x ≡ 5(mod 8), x ≡ 3(mod 18)
b x ≡ 2(mod 3), x ≡ 5(mod 6), x ≡ 7(mod 8) c
x ≡ 3(mod 5), x ≡ 4(mod 6), x ≡ 4(mod 8)
d x ≡ 5(mod 8), x ≡ 3(mod 7), x ≡ 1(mod 9) Find the quadratic residues for: a
E
15
modulo 6
b modulo 11 modulo 15
d modulo 20
PL
c
16
Show that 3 is a quadratic residue of 23 and is a non-residue of 31.
17
Solve the following congruences: a
x 2 ≡ 4(mod 9)
b x 2 ≡ 6(mod 10) x 2 ≡ 11(mod 14)
SA M
c
d x 2 ≡ 7(mod 18)
If x 2 ≡ q(mod n), where n > 2, has one solution x ≡ x 0(mod n), state another solution in terms of n and x 0.
18
Prove that the following have no solutions: a
x 2 ≡ 5(mod 9)
b x 2 ≡ 3(mod 7) c
x 2 ≡ 2(mod 16)
d x 2 ≡ 3(mod 19)
19
List all the quadratic residues and all the quadratic non-residues for a
modulo 5
b modulo 7 c
modulo 11
d modulo 13 State the number of quadratic residues and non-residues for modulo p where p is prime. file:///C:/Profiles/j.raja/Desktop/New%20folder%20(2)/New%20folder%20(2)/P1.html
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