What is the last digit that is the units digit of 2 to the p

What is the last digit (that is the units digit) of 2 to the power of 10^6? What number between 121 and 149 is exactly divisible by both 3 and 5? (Show the MATLAB code used.) A standard clock is set correctly at 1:00 PM. If it loses 3 minutes every hour, what will the clock show when the correct time is 10:00 AM the next day? (Show the MATLAB code used.) The six-digit number 63X90Y is an even multiple of 27. What digits do X and Y represent? (Show the MATLAB code used.)

Solution

SOLUTION:

7.

10 = 4.2+2

10² = 100 = 4.24+4 = 4.24+2²

......................................

10⁶ = 4.k+2⁶

   = 4.k\' for some natural number k. (since, 2⁶ =64 = 4.16 so, k^\'=k+16)

Now, last digit of 2⁴ is 6.

Therfore, last digit of 2^4k\' is 6^k\'= 6.

That is, last digit of 2 to the power 10⁶ is 6.

8.

Let m be the desired number between 121 and 149.Then,

3 divide m and 5 divide m. Therefore 3.5 divide m.That is 15 divide m.

So the posiible choices for m is m = 15k , k is positive integer and 121<m<149.

Take k= 9.Then m=135 and 121<135<149. Also 3divide 135 and 5 divide 135.

So the desired number is 135.