KMB, 02/11/99

M3/4P14 Elementary Number Theory—Quick fire test.

1) True or false? (just say T or F)

a) If a ≡ 1 mod b and b ≡ 1 mod c then a ≡ 1 mod c.

b) If a ≡ 1 mod c and b ≡ 1 mod c then ab ≡ 1 mod c.

c) if a ≡ b mod m and c ≡ d mod n then a + c ≡ b + d mod m + n.

d) if a ≡ b mod m and c ≡ d mod n then ac ≡ bd mod hcf(m, n).

e) if a ≡ 0 mod m and b ≡ 0 mod m then ab ≡ 0 mod m^{2}.

2) I am given that x is an integer such that x ≡ 6 mod 10. What can I say about x mod 2, mod 3, mod 4, mod 20, mod 30, and mod 31?

3) True or False?

a) If 4x ≡ 4y mod 27, then x ≡ y mod 27.

b) If 4x ≡ 4y mod 28 then x ≡ y mod 28.

c) If 4x ≡ 4y mod 30 then 2x ≡ 2y mod 30.

4) Fill in the three “?”s with numbers that make the statements true.

a) 6x ≡ 6y mod 25 if and only if x ≡ y mod ?

b) 6x ≡ 6y mod 24 if and only if x ≡ y mod ?

c) 6x ≡ 6y mod 26 if and only if x ≡ y mod ?

5) If you’ve done all those, then whilst you’re waiting for the others, try this: Let p ≥ 5 be a prime, and set n = \binom{2p}{p}.

(i) Is it true that n ≡ 2 mod p? (ii) Is it true that n ≡ 2 mod p^{2}? (iii)^{†} Is it true that n ≡ 2 mod p^{3}?