Question

The following systems of congruences do not have pairwise coprime moduli. Determine in each case if the system has a solution and, if it does, determine the congruence class consisting of all its solutions. If it does not have a solution, explain why. (i) { x≡5mod24 (ii) x=1 , mode6

x≡25mod36 x=5, mode8
​ x=4, mode9

Answer 1

The systems of congruences do not have pairwise coprime moduli. (i) does not have a solution. (ii) has a solution x≡37(mod72).

Step-by-step explanation:

The systems of congruences (i) and (ii) do not have pairwise coprime moduli, so we need to find the least common multiple (LCM) of the moduli to determine if there is a solution.

(i) { x≡5mod24 (ii) x=1 , mode6 x≡25mod36 x=5, mode8​ x=4, mode9

(i) To find LCM(24), we need to factorize 24 into its prime factors: 24 = 2^3 * 3. Since 5 is not divisible by 2 or 3, the system does not have a solution.

(ii) To find LCM of 6, 36, 8, and 9, we need to factorize each number: 6 = 2 * 3, 36 =
`2^2 * 3^2, 8 = 2^3, and 9 = 3^2.`hest power of each prime factor: LCM(6, 36, 8, 9) =
`2^3 * 3^2`nce x=1(mod6) and x=5(mod8) are both divisible by 2, and x=1(mod6) and x=4(mod9) are both divisible by 3, there is a solution. The congruence class consisting of all solutions is x≡37(mod72).