Question

Find the Unit digit of 7^125

Answer 1

see, there is a rule called ' universal cyclicity ' of exponents

lets take an example, observe the pattern below :-

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6= 729

we can observe that the last digit repeats itself after a cycle of 4 and the cycle is 3, 9, 7, 1 this repetition of numbers after a particular stage is called the cyclicity of numbers.

Therefore when we need to find the unit digit of any number like 3^n we just need to find the number on which the cycle halts. So we divide power n by 4 to check remainder

so, according to the question  we need to find the unit digit of 7^125

we can use the concept of cyclicity of 7 here,

1. If remainder is 1 then the unit digit will be 7
2. If remainder is 2 then the unit digit will be 9
3. If remainder is 3 then the unit digit will be 3
4. If remainder is 0 then the unit digit will be 1

so, if we divide 125 with 4 we get 1 as the remainder

hence, according to the cyclicity of of seven the unit ndigit of 7^125 will 7

Hope it helps :)