TOUGH NUMBER THEORY QUESTIONS

Vidushi Jain wrote:What is the last digit of the number 23457^194321=
A. 9
B. 1
C. 3
D. 7
E. 34

Let us examine the last digits when 7 is raised by successive integers to see if we can find a pattern

7^1 -> 7
7^2 -> 9
7^3 -> 3
7^4 -> 1
7^5 -> 7 .... So the numbers repeat every time 7 is raised.

Pattern of last digits 7 9 3 1 7 9 3 1 7 9 3 .....

If we divide 194321 by 4 we get 1 as remainder. So we need to find the last digit of 7^1 = 7.
Hence D.

Vidushi Jain wrote:What is the last digit of the number 23457^194321=
A. 9
B. 1
C. 3
D. 7
E. 34

Since we know the last digit of the square of a certain number is determined by the last
digit of that number itself. We can get the cycle that
the last digit of 7^1=>1, 7^2=>9, 7^3=>3 and 7^4=>1....
So we have to make sure at what point 194321 is in the cycle
Since we can tell that 194320 is divisible by 4. So 194321 lies on the first in the cycle
So the answer is D