# mod

1. ### Show that x^2 is congruent to 1 (mod 24) given x congruent to 5 (mod 6)

Hi all, I will write out the word congruent instead of the symbol, as we use three lines over one another and I could not find a way to write it out. The question: "Let x be congruent to 5 (mod 6). Show that x^2 is congruent to 1 (mod 24)." I believe I need to set up one equation as indices...
2. ### how to prove multiplicative integers (mod p) has at least one generator?

How do you prove the multiplicative integers (mod p) has at least one generator? I know how to show a^(p-1) is congruent to 1 using Fermat's Little Theorem but how do you show p-1 is the smallest such integer? Thanks!
3. ### Proof: If q is a prime divisor of 2^p -1, then q ≡ 1 (mod p)

Let p be a prime number and M = 2^p − 1. Let q be a prime divisor of M. Prove that q ≡ 1 (mod p). I'm completely stuck. Help please?
4. ### Show that n=1 (mod 9)

Show that $n\equiv 1 \pmod 9$ for all even perfect numbers $n>6$.
5. ### mod and div questions

Hi , Sorry for posting simple questions but I don't know how to solve these questions : Q1 : The smallest positive integer a such that a + 1 ≡ 2a (mod 11) is ...ANSWER IS 12 Q2 : Find –151 div 20 ANSWER IS -8 Q3 : Find –151 mod 20 ANSWER IS 9 thanks a lot.
6. ### Find an inverse for (x^2+2x+1) (mod x^3 +x^2+1) in Z_3

So I have (x^2+2x+1)q(x) \equiv 1 \mod{x^3 +x^2+1} over \mathbb{Z}_3 And the goal is to find $q(x)$ I'm working off a partial solution and I'm trying to fill in the holes at the moment. So firstly we check if the modulus is reducible/factorable, since it has a root $(f(x)=1+1+1=3=0)$ it is...
7. ### Find a multiplicative inverse of a polynomial mod (x^2-2)

(a+bx)q(x) \equiv 1 \mod {x^2-2} Since we're concerned only with remainders and the modulus is second order, $q(x)$ must be linear. So we set $q(x)=(c+dx)$ (a+bx)(c+dx) \equiv 1 \mod {x^2-2} ac+adx+bxc+bdx^2 \equiv 1 \mod {x^2-2} Substituting $x^2=2$ and rewriting gives, (ac+2bd)+(ad+bc)x...
8. ### Find the least residue of the multiplicative inverse of a mod m

A) a=7, m=11 B)a=7, m= 23 C)a=5, m=31 D)a=117, m=24
9. ### integration of mod x - regarding

kindly help me how to integrate mod x ? there is no limits for the integration; indefinite integral. with warm regards, ARANGA
10. ### Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.

Im having trouble with this proof. Could someone help me? Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.
11. ### show (m + dk) mod d = m mod d

I just learned about mod, I know that for example 10 mod 3 = 1, 1 being the remainder of 10 divided by 3 we just covered induction but I don't feel this is a problem that can be solved by induction, or not in what I have learnt thus far question is: If m, d, k \mathhbb{Z} and d>0 then prove...
12. ### mod 7

find the smallest positive integer b which satisfies 3^56=b(mod7) I am very new to modular arithmetic and I stubbled upon this question. Wouldn't 'b' have a specific solution. Asking for the smallest positive integer b confuses me. 3^{56} - b = 7k \;\;\text{ Where k is an integer} when...
13. ### Calculation in Rijndael field GF(2^8) and (mod x^4+1)?

I have a 6th degree polynomial with unknown coefficients, but I need to take it (mod x^4+1) to find some 3rd degree polynomial. Using wolfram alpha and mathematica 9.0, I could only reduce the polynomial to a different 6th degree polynomial that was shorter. e.g. a_1x^6 + a_2x^5 + ... a_6x...
14. ### Prove that σ(n)≡d(m) (mod 2) where m is the largest odd factor of n.

Prove that σ(n)≡d(m) (mod 2) where m is the largest odd factor of n. need elaborate and quick reply please.
15. ### For which n is 1^2 + 2^2 + ... + (n-1)^2 ≡ 0 mod n true?

For which n ≥ 2 is the following true; 1^2 + 2^2 + ... + (n-1)^2 ≡ 0 mod n I've tried some numbers and found that it works for 5 but I'm not exactly sure why that is. I'm aware of that 1^2 + 2^2 + ... + (n-1)^2 is the sum of every element in complete residue system for n multiplicative by...
16. ### prove there is no primitive root mod 3p

when p odd , p>3. --------- I don't know how to cintinue: a^(p-1)=1(modp) a^2=1(mod3) ....
17. ### solving for x in 78^x =x (mod 10^12)

Hi Guys, Any idea how to solve the equation 78^x = x (mod 10^12) ? Assume that x is a 12 digits number I've looked over every congruence and can't seem to find anything useful.
18. ### Show a^n = 1 (mod 2^n+2) if a odd.

I know I have to use induction. The n=1 is simple: a^2 = (2j+1)^2 = 4j^2 + 4j + 1 = 1 mod 8 = 1 mod 2^3 Then we assume for a^n that the claim is true. However I'm not sure on the inductive step. This is a HW assignment and help would be much appreciated. Thanks!

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