Modular Arithmetic
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Simplifying large exponents in modular arithmetic like
WEBSep 12, 2013 — I was simplifying a larger modular arithmetic problem ($2013^{2014} \pmod{5}$) and got it down to $4^{1007} \pmod{5}$ and am wondering if there's a general approach to dealing with large exponents like $1007$. In general, what approaches are there to simplify large exponents like $1007$ when doing modular arithmetic?
Math.stackexchange.comModular Arithmetic - Find the Square Root - Mathematics Stack …
WEBWe should know that 25 has two square roots in ordinary arithmetic: 5 and -5. modular arithmetic SQUARE ROOTS. IF the square root exists, there are 2 of them modulo a prime. To continue our example, 25 has the two square roots 5 and -5. We can check this: (− 5)2 = 25 ≡ 3 mod 11 (5)2 = 25 ≡ 3 mod 11. To find the square roots sometimes
Math.stackexchange.commodular arithmetic - How do you calculate the modulo of a high …
WEBI need some help with this problem: $$439^{233} \\mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks.
Math.stackexchange.commodular arithmetic - How can I find a mod with negative number
WEBTherefore 31 = 7 ⋅ 4 + some number 31 = 7 ⋅ 4 + some number, where your goal is to determine what some number some number is. This same exact process applies for negative numbers. If you want to evaluate −11 (mod 7) − 11 (mod 7), you need the largest multiple of 7 7 that's less than or equal to −11 − 11. This is −14 − 14.
Math.stackexchange.comNewest 'modular-arithmetic' Questions - Mathematics Stack …
WEBmodular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation
Math.stackexchange.comHow to find the inverse modulo - Mathematics Stack Exchange
WEB$ $ The calculations are usually simpler using modular fraction arithmetic e.g. see here, and here and here for circa $20$ motley worked examples via a handful of methods (and see the sidebar "Linked" questions lists there for many more).
Math.stackexchange.commodular arithmetic - Last digits, numbers - Mathematics Stack …
WEB2) Find the last 2 digits of 332014. Attempt: 1) By just setting the powers of 7 we have 71 = 7, 72 = 49, 73 = 343, 74 = 2401, 75 = 16807, 76 = 117649, …. After the power of 4, the last digits will repeat. Then by noticing the pattern the digits will end in 7, 9, 3 and 1. Then we can divide the exponent (12345) by 4 since this is the cycle
Math.stackexchange.commodular arithmetic - What are the properties of the modulus
WEB3. In mathematics "modulus" % is not a well-defined arithmetic operation. At best it might be considered a function that returns an equivalence class rather than a number. Mathematics treats "mod" as an equivalence relation. Even in a computer programming context you can easily verify the operation is not commutative, e.g. 2 % 4 is not 4 % 2.
Math.stackexchange.comHow to do a modular arithmetic with negative exponents?
WEB1. IF (and it is a big if) gcd(a, n) = 1 gcd ( a, n) = 1 then we know that there a power k k ( ϕ(n) ϕ ( n) is one such possible k k) so that ak ≡ 1 mod n a k ≡ 1 mod n and that and the aj a j form a group under multiplication modulo n n. And further that aj+k ≡aj mod 25 a j + k ≡ a j mod 25. [It does NOT follow that k = ϕ(n) k = ϕ
Math.stackexchange.comModular Arithmetic: using congruence to find remainder
WEB2. First, the remainder when blah is divided by 7 7 means the congruence of blah mod 7 7. The significance of the result you quoted is that when doing a computation in which you’re only interested in the final congruence modulo 7 7, you can always replace a number by any other number that’s congruent to it. So, 32 3 2 may be 9 9, but you
Math.stackexchange.com