-2
$\begingroup$

X is a number formed by writing 9 for 99 times. What will be the remainder of this number when divided by 7?

$\endgroup$
  • 2
    $\begingroup$ So you're looking at $10^n -1 \mod{7}$. Maybe you can give some values for $n$ and see if you find a pattern. $\endgroup$ – Matti P. Mar 19 at 14:04
  • 1
    $\begingroup$ Is there anything you have already tried? I can think of two methods which enable me to do this in my head, but just telling you what they are won't help you much in solving similar problems - trying it yourself will help. $\endgroup$ – Mark Bennet Mar 19 at 14:05
  • $\begingroup$ Are you familiar with Fermat's little theorem? $\endgroup$ – J. W. Tanner Mar 19 at 14:06
1
$\begingroup$

$X= 10^{99} - 1 = 10^{3 \times 33} - 1$.

Now $10^3 = 1000 = -1$ modulo $7$.

So $X=(-1)^{33} - 1 = - 1 - 1 = 5$ modulo $7$.

$\endgroup$
  • $\begingroup$ Easier: $\bmod 7\!:\,\ 10^{\large 3}\equiv 3^{\large 3}\equiv 3\cdot 2 \equiv -1\ \ $ $\endgroup$ – Bill Dubuque Mar 19 at 15:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.