# lest significant digit

• Jan 7th 2009, 03:06 PM
pablo26
lest significant digit
Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
eg 1002^3755
• Jan 8th 2009, 12:40 AM
Constatine11
Quote:

Originally Posted by pablo26
Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
eg 1002^3755

The least significant digit of $N$ is $N \pmod {10}$.

So in the case of your example:

$1002^{3755}\equiv 2^{3755} \pmod{10}$

Now (you should know this) $2^{10}=1024$
so:

$
2^{3755}=(2^{10})^{375}2^5\equiv 4^{375} \times 2 \pmod{10}
$

$
4^{375} \times 2=2^{751}=(2^{10})^{75} \times 2 \equiv 4^{75} \times 2 \pmod{10}
$

$
4^{75} \times 2=2^{151}=(2^{10})^{15} \times 2 \equiv 4^{15} \times 2 \pmod{10}
$

$
4^{15} \times 2=2^{31}=(2^{10})^3 \times 2\equiv 4^3 \times 2 \pmod{10}
$

and:

$
4^3 \times 2=128 \equiv 8 \pmod{10}
$

Hence:

$1002^{3755}\equiv 8 \pmod{10}$

and so the least significant digit of $1002^{3755}$ is $8$.

.