Compute the last digit of 601201620 in base 10 using only mo

Compute the last digit of (6012016)^20 in base 10 using only modular arithmetic (not via calculator). Justify and show each step you take.

Solution

for 6012016 ¹ --> the units digit is 6 (mod 10)

for 6012016 ²--> the units digit is 6 (mod 10)

for 6012016 ⁴--> the units digit is 6. (mod 10)

for 6012016 ⁸--> the units digit is 6. (mod 10)

for 6012016 ¹⁶--> the units digit is 6. (mod 10)

for 6012016 ¹⁶* 6012016 ^4  -->--> the units digit is 6. (mod 10)

as we can observe that for every power of the given number the units digit is same, so for the power of 20 for the number 6012016 the last digit is \"6\".

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(6012016)²⁰ =6012016¹⁰ * 6012016⁵ *6012016² *6012016² *6012016¹

last digits = 6*6*6*6*6

= 6

for (6012016)² units digit is 6 and again the units digit of square of the obtained number is 6.

and for every number which is ends with 6, the last digit of the power of that number is \"6\" irrespective of power value.