what is the ones digit?

Question

What is the ones digit of 11^11+14^14+16^16? I also have to use the strategy "Solve a simpler problem". Can someone please help. Thanks!

Answer 1

With n>0:

 

11ⁿ will always have a 1's digit of 1

11¹¹ has 1's digit of 1

 

14ⁿ will have 1's digit of 4 if n is odd

14ⁿ will have 1's digit of 6 if n is even

14¹⁴ has 1's digit of 6

 

16ⁿ will always have a 1's digit of 6

16¹⁶ has 1's digit of 6

 

1+6+6 = 13

The '1' from the 10's space will be carried in

your addition leaving 3 in the 1's digit space 

of the sum.

 

Answer 2

The strategy "the strategy 'Solve a simpler problem'" means that you identify sub-problems, solve them, then put those solutions together in order to solve the big problem.

 

This math problem seems to be: A + B + C

 

Problem A:  What is 11^11 ??

 

    This expands to:  What is 11*11*11*11*11*11*11*11*11*!1*11?

     11*11 = 121

     Then 121 * 11 and the following partial sums (the running product) may be written as:

            xxxxx1      [note: x is some digit]

                * 11

         ------------

           xxxxxx1             [note: the ones digit is always 1]

                 * 11

         ------------

           xxxxxxx1           [note: the ones digit is always 1]

 

              . . .             [do this 11 times]

--------------------

      xxxxxxxxxx1          [So, the ones digit for Sub-problem A is 1]

 

 

Now, the ones digit (the right-most digit) for Sub-problem C is always 6 [PLZ check this yourself].

 

But, watch out!  The ones digit for Sub-problem B alternates between 6 (for even exponent) and 4 (for odd exponent),

so you must select the correct value, then add the ones digits for Sub-Problems A+B+C.

 

NOTE:  The problem only asks for the ones digit of the grand sum (A+B+C).