modulo

[PURE MATH]

1. (SSSMO/1997) Find the smallest positive integer n such that $1000 \leqslant
n \leqslant 1100$ and $1111^n + 1222^n + 1333^n + 1444^n$ is divisible by 10
2. Find the remainder of $47^{37^27}$ when it is divided by 11.
3. (SSSMO(J)/2001) Find the smallest positive integer k such that $2^69+
k$ is divisible by 127.

[oKiNeSIuMo]

1. 1001
2. 5
3. 63

[lnพwsะบุ๑sสุ๑xล่o]

1. (SSSMO/1997) Find the smallest positive integer n such that $1000 \leqslant
n \leqslant 1100$ and $1111^n + 1222^n + 1333^n + 1444^n$ is divisible by 10
2. Find the remainder of $47^{37^{27}}$ when it is divided by 11.
3. (SSSMO(J)/2001) Find the smallest positive integer k such that $2^{69}+
k$ is divisible by 127

hint1

เช็คหลักท้าย ทำได้หลายวิธี

hint2

$47^{10}\equiv 10(mod11)$ & $37\equiv 10(mod11)$

hint3

$2^7\equiv1(mod127)$ , เศษ 127-k