Units place in math expression

Many exams try to make you feel that you cannot solve the problem by making question as big as possible like units place of (186937)^178909 but solving this problem is as simple as solving 9%4

Before solving these kind of problems, understand the great concept called Cyclicity of numbers which states that any digit when multiplied by itself repeatedly, generates a cycle of digits.

points to remember:

• Only units place will be responsible for generating units place
• Every number after applying successive multiplications,repeats the cycle of digits.
  
  
  
  
  
  
  

So try to figure out a common series in powers of digits

for example units(3)=3,units(3*3)=9,units(9*3)=7,units(7*3)=1,units(1*3)=3(we get a cycle of multiplication)

Number      powers                         Series/cycle(units place of powers column) [cyclicity of number]

2                 2,4,8,16,32                   2,4,8,6                    [4]
3                 3,9,27,81,243               3,9,7,1                    [4]

4                 4,16,64                         4,6                          [2]
7                 7,49,343,2401              7,9,3,1                    [4]
8                 8,64,512,4096              8,4,2,6                    [4]
9                 9,81,729                       9,1                          [2]

Numbers 0,1,5,6 generates same numbers when multiplied y themselves, so the cyclicity of numbers for 0,1,5,6 is [1] .
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Algorithm for finding units place

1. Reframe the entire number by discarding all digits except units place.
2. Find the cycle count of the units place.
3. find the remainder when power is divided by cycle count(Exponent%cyclecount) =R
4. R th number in the cycle is the answer.

Example 1:What is the units place of (186937)^178909
solution:
As units place alone generates units place,this is equivalent to units place of (7)^178909
Cyclicity of number 4 is 4,
R=178909%4=1
1st number of cycle is 7.

Example 2: what is the units place of ((188888888888888)*18888888888888888888885)^19979
solution:
5*8=40
units place=0
The given question can be reframed as (0)^19979
Answer is 0.

Example 3:what is the units place of 129*189*123*1773*87127*999
Solution:
As units place alone is responsible to generate units place,
answer = 9*9*3*3*7*9
            =81 * 9 * 63 [again take units place]
            = 1* 9* 3
            =27
units place = 3