Question

Find the unit digit of number
3^27542 x 4^25493 + 6^275485×
5^12547
​

Answer 1

Answer:

        6

Step-by-step explanation:

The units digit of powers of 3 cycle every 4:

 3¹ = 3,  3² = 9,  3³ = ...7,  3⁴ = ...1,  3⁵ = ...3, etc.

Since 27542 leaves a remainder of 2 when divided by 4, the units digit of 3²⁷⁵⁴² is 9.  [ That is, raising to the power 27542 cycles the units digit through 3 - 9 - 7 - 1 over and over, finishing at the second number. ]

Similarly, the units digit of powers of 4 cycles every 2:

 4¹ = 4,  4² = ...6,  4³ = ...4, etc.

Since 25493 leaves a remainder of 1 when divided by 2, the units digit of 4²⁵⁴⁹³ is 4.

The units digit of powers of 6 is always 6, and the units digit of powers of 5 is always 5.

So the units digit of

   3²⁷⁵⁴² × 4²⁵⁴⁹³ + 6²⁷⁵⁴⁸⁵ × 5¹²⁵⁴⁷

is the same as the units digit of

  9 × 4 + 6 × 5 = 36 + 30 = 66

so the answer is 6.