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.
- HI There! By the way, you can use the latex command \neq to made the symbol "not equal to". In fact, using latex, your post could look like this
- $ \begin{align} 2\times 2 &= 4 \\ \text{ but } 4\text{ mod } 10 &= 4 \neq 2. \end{align} $
- Thus 2 cannot be a unity.
- $ \begin{align} 4\times 4 &=16 \\ \text{but } 16 mod 10 &= 6 \neq 4. \end{align} $
- Thus 4 cannot be a unity.
- $ \begin{align} 6\times 6 &= 36 \\ \text{and } 36\text{ mod }10 &= 6. \text{ (Check) } \end{align} $
- But this is not enough. We must check what effect multiplying 6 has on the other elements.
- $ \begin{align} 0\times 6 &= 0 \text{ and } 0 \text{ mod } 10 = 0. \text{ (Check) } \\ 2\times 6 &= 12 \text{ and } 12 \text{ mod } 10 = 2. \text{ (Check) }\\ 4\times 6 &= 24 \text{ and } 24 \text{ mod } 10 = 4. \text{ (Check) }\\ 8\times 6 &= 48 \text{ and } 48 \text{ mod } 10 = 8. \text{ (Check) }\\ \end{align} $
--Bakey 08:45, 10 October 2012 (UTC)
