Answer by lab bhattacharjee for How do I find the remainder for the following?
For $19^{a+1}\pmod{38},$let us find $19^a\pmod{\dfrac{38}{19}}$As $19\equiv1\pmod2$$19^a\equiv1^a\equiv1\pmod2$ for integer $a\ge0$$\implies19^{a+1}\equiv1\cdot19\pmod{2\cdot19}$See also : How to find...
View ArticleAnswer by Bill Dubuque for How do I find the remainder for the following?
$19(2)\mid 19(18)\mid 19^{n+1}\!-19$since: $\, a(a\!-\!1)\mid a(a^n-\,1)$
View ArticleAnswer by Ana Paula Chaves for How do I find the remainder for the following?
Precisely, Euler's totient function comes to be very handy for such problems. Euler's theorem states that if gdc$(a,m)=1$, then $a^{\phi(m)} \equiv 1 \pmod{m}$. In your case, since gdc$(19,38) \neq 1$,...
View ArticleAnswer by wonderman for How do I find the remainder for the following?
Since $19^{38} - 19$ is divisible by both 2 and 19, it is divisible by 38. Therefore,$$19^{38} \equiv 19 \pmod{38}.$$Euler's Theorem can't be applied here, since $\gcd(19,38)$ is not equal to 1. But...
View ArticleHow do I find the remainder for the following?
I know this is a very typical question for modular arithmetic but still I haven't found a comprehensive explanation for this question, so I'm posting it here. So here goes:I need to find the remainder...
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