The problem is giving you that the {0, 2 , 4 , 6 ,8} under addition and multiplication modulo 10 has a unity, and wants you to find it.
By definition we know the unity of a ring is a nonzero element that is an identity under multiplication.
Furthermore, if we consider each element of {0, 2 ,4 , 6, 8} (other than zero) we can see which element has this property.
Note: Let the symbol '<>' be equivalent to "not equal to"
2*2= 4 but 4 mod 10 = 4 <> 2. Thus 2 cannot be a unity. 4*4 =16 but 16 mod 10 = 6 <> 4. Thus 4 cannot be a unity. 6*6 = 36 and 36 mod 10 = 6. (Check) But this is not enough. We must check what effect multiplying 6 has on the other elements. 0*6 = 0 and 0 mod 10 = 0. (Check) 2*6 = 12 and 12 mod 10 = 2. (Check) 4*6 = 24 and 24 mod 10 = 4. (Check) 8*6 = 48 and 48 mod 10 = 8. (Check)
Therefore 6 must be the unity.
--Bakey 08:45, 10 October 2012 (UTC)
