Unit Digit of A Simplified Fraction If $a$ is an integer such that $a=xyz4$, where $x,y,z$ are unit digits, and $b$ is a positive integer such that $b=mnp6$, where $m,n,p$ are unit digits, and $a$ and $b$ are not coprime, what are some possible values for the units digit in the numerator and the denominator after $a/b$ has been simplified. 
$a/b=dfg2/hjk3$ is one possibility but it's not guaranteed. 
 A: You have to find out the units digit.  
See, that when units digits are concerned, you have only a limited number of possible values during multiplication. If you want to get $a_1...a_n4$ as when you multiply $b_1...b_m3$ by $c_1...c_lx$, then $x$ will always invariably be 8 $(3×8=24)$.
So, the whole operation will be based off the unit digits.
Case 1: If the Numerator is Completely Divisible by the Denominator
$Given,$
${xyz4}\over{mnp6}$
$={{xyz4}\over{6}}$
$Since 6×4=24 & 6×9=54$
$So,$
$\text{The units digit is }4\text{ OR }9$
Case 2: If the Numerator is not Completely Divisible by the Denominator
Given that the a and b are not coprime, they must have a GCD (Which can't be odd since then, they would be even again, and then would have 2 as a common divisor, which would negate the suggestion it's the GCD). Since the digits are 4 and 6, this must have multiple which end with both of those. So let's see those numbers, case by case.
Case 2.1
$GCD=x_1...x_n2$
So, by the above process, the number will be ${x_1...x_m2}\over {y_1... y_n3}$
And the unit digits will be$2\over 3$
Case 2.2
$GCD=x_1...x_n4$
So, the number will be: 
${x_1...x_m1}\over{y_1... y_n4}$
$OR$
${x_1...x_m1}\over{y_1... y_n9}$
$OR$
${x_1...x_m6}\over{y_1... y_n9}$
And the unit digits will be $1\over 4$ or $1\over9$ or $6\over9$
Case 2.3
$GCD=x_1...x_n6$
So, the number will be: 
${x_1...x_m4}\over{y_1... y_n1}$
$OR$
${x_1...x_m9}\over{y_1... y_n1}$
$OR$
${x_1...x_m9}\over{y_1... y_n6}$
And the unit digits will be $4\over 1$ or $9\over1$ or $9\over6$
Case 2.4
$GCD=x_1...x_n8$
So, the number will be: 
${x_1...x_m3}\over{y_1... y_n2}$
$OR$
${x_1...x_m3}\over{y_1... y_n7}$
$OR$
${x_1...x_m8}\over{y_1... y_n7}$
And the unit digits will be $3\over 2$ or $3\over7$ or $8\over7$
So,
The possible digits are: 
Case 1
$4,9$
Case 2
$2\over3$,$1\over4$,$1\over9$,$6\over9$,$4\over1$,$3\over2$,$3\over7$,$8\over7$
