what is the last 2 digits of 3 raised to 1994 and 7 raised to 1994??
Plz help me this is urgent
Hello, xXxSANJIxXx!
I found two different methods for these problems . . .
What are the last 2 digits of ?
Since 3^2)^{997} \:=\:9^{997} \:=\10 - 1)^{997}" alt="3^{1994} \:=\3^2)^{997} \:=\:9^{997} \:=\10 - 1)^{997}" />, consider the binomial expansion:
. .
. . . .
. . . . . . . . . These do not affect the last two digits
Therefore, the last two digits are: .
What are the last 2 digits of ?
We note that:
. . Hence: will end in .
Since , we have: .
. . . . .
. . (a number ending in )
Therefore, the last two digits of are:
I have a different approach. It involves the phi function.Originally Posted by xXxSANJIxXx
Since,
you have,
To find, prime factorize 100,
Thus,
Thus,
Raise both sides to the 50th power,
Express as,
Let, then,
Solve for ,