a)Which of the residues $0, 1, 2, \ldots, 11$ satisfy the congruence $3x \equiv 0 \pmod{12}?$Give your answer as a list, separated by commas, in order from least to greatest.
b)Which of the residues $0, 1, 2, 3, 4$ satisfy the congruence $x^5 \equiv x \pmod{5}?$Give your answer as a list, separated by commas, in order from least to greatest.
c)Which of the residues $0, 1, 2, 3, 4, 5$ satisfy the congruence $x^2 + 3x + 2 \equiv 0 \pmod{6}?$Give your answer as a list, separated by commas, in order from least to greatest.
+118723 a)Which of the residues \( 0, 1, 2, \ldots, 11\)
satisfy the congruence \(3x \equiv 0 \pmod{12}?\)
Give your answer as a list, separated by commas, in order from least to greatest.
I am not sure what you mean by reidues??
\(3x \equiv 0 \pmod{12}?\)
Only x=0 and x= multiples of 4 would make this true. So x=0,4,8
b)Which of the residues $0, 1, 2, 3, 4$ satisfy the congruence \(x^5 \equiv x \pmod{5}? \)$Give your answer as a list, separated by commas, in order from least to greatest.
0^0=0 satisfies
1^5=1 satisfies
2^5=32=2 mod5 satisfies
3^5=243= 3mod5 satisfies
4^5=1024=4mod5 satisfies
Nothing bigger will satisfy.
x=0,1,2,3,4 will all satisfy this equation.
c)Which of the residues $0, 1, 2, 3, 4, 5$ satisfy the congruence
\(x^2 + 3x + 2 \equiv 0 \pmod{6}? \)
Give your answer as a list, separated by commas, in order from least to greatest.
\(x^2 + 3x + 2 \equiv 0 \pmod{6}?\\ If \;x=0\\ 0 + 0 + 2 \equiv 2 \pmod{6}?\qquad \mbox{Does not satisfy}\\ If \;x=1\\ 1 + 3+ 2 \equiv 0 \pmod{6}?\qquad \mbox{DOES satisfy}\\ If \;x=2\\ 4 + 6 + 2=12 \equiv 0 \pmod{6}?\qquad \mbox{DOES satisfy}\\ If \;x=3\\ 9 + 9 + 2=20 \equiv 2 \pmod{6}?\qquad \mbox{Does not satisfy}\\ If \;x=4\\ 16 + 12 + 2=30 \equiv 0 \pmod{6}?\qquad \mbox{DOES satisfy}\\ If \;x=5\\ 25 + 15 + 2=42 \equiv 0 \pmod{6}?\qquad \mbox{DOES satisfy}\\\)
So x=1,2,4,5 satisfy the equality.
You should check this. If you need more explanation then ask for it :)
