What is the $10^5\pmod{35}$?

Question

How to evaluate $10^5 \pmod {35}$ ?

I tried this $a=(10^2\cdot 10^3)\pmod{35}$
then again a mod $35$. This is very lengthy please tell me a shorter way?

Easy to do this directly. If you prefer, you can do it $\pmod 5$ (trivial) and $\pmod 7$ separately and then use the Chinese Remainder Theorem. — lulu, May 15 '18 at 15:32

score 2 · Accepted Answer · answered May 15 '18 at 15:41

2

I wouldn't say "lengthy". You have $10^2=100\equiv30$, so $$10^5=10^210^210\equiv30\times 30\times10\equiv 30\times300\equiv30\times 20\equiv(-5)(-15)\equiv75\equiv 5. $$ Either the above, or just do long division 10000 in 35.

answered May 15 '18 at 15:41

Martin Argerami

205,756

score 1 · Answer 2 · answered May 15 '18 at 15:35

1

HINT

Note that

$$10^2 \equiv100\equiv-5\pmod{35}$$

and

$$10^5 \equiv 10^2\cdot 10^2\cdot 10\equiv-5\cdot -5 \cdot 10\pmod{35}$$

answered May 15 '18 at 15:35

user

154,566

score 0 · Answer 3 · answered May 15 '18 at 15:35

$$ 10^2 \equiv 30 \left[35\right] \Rightarrow 10^3 \equiv 300 \left[35\right]\equiv 20 \left[35\right] $$ So $$ 10^4 \equiv 200 \left[35\right] \equiv 25 \left[35\right] \Rightarrow 10^5 \equiv 250 \left[35\right] $$ So

$$ 10^5 \equiv 5 \left[35\right] $$ ( and in fact $ 10^5=2857*35 +5$ )

score 0 · Answer 4 · answered May 18 '18 at 13:31

As $(10^5,35)=5$

let us find $10^{5-1}\mod\left({\dfrac{35}5}\right)$

Now $10\equiv3\pmod7,10^4\equiv3^4\equiv4$

$\implies10^5\equiv4\cdot10\pmod{7\cdot10}\equiv40\pmod{35}\equiv?$

See also:

How to find last two digits of $2^{2016}$

Get the last two digits of $16^{100}$ and $17^{100}$

What is the $10^5\pmod{35}$?

4 Answers4