Find an example of the following cf Exercise 150 ab equivale

Find an example of the following (cf. Exercise 1.50). ab equivalence 0 (mod n) but a, b not equivalence 0. ab equivalence ac (mod n) with a 0 and b not equivalence c. a^2 = b^2 (mod n) but a not equivalence plusminus b.

Solution

Ans a: Let a = 9; b=4 and n=18; ab= 36 and 36=0 (mod 18) since 36 is divisible by 18. but 9!=0 (mod 18) and 4!=0 (mode 18) since both 9 and 4 are not divisible by 18;

Ans b: Let a=2; b=3 and c=6 and n=6; ab=6 and ac=12; 6 (mod 6)= 12 (mod 6)= 0 thus ab=ac(mod n)
But a(mod n) = 2 (mod 6)=2 which is not 0; and mod b= 3 and mod c=0 (with respect to n=6) so b != c
Note: != stands for is not equal to

Ans c: let a=3 and b=7 and n=20; a^2= 9 and b^2=49; 9=49 (mod 20) since both leave a remainder of 9 when divided by 20; but mod a= 3 (with respect to n=20) and mod +b= mod7 and mod -b= mod -7 none of which are equal to 3 with respect to n=20;