Question

Define f:Z→Z_m × Z_n :f (x)=(xmodm,xmod n),m,n∈N.

i) If (m, n) = (3, 4), find Ker f.

ii) If (m, n) = (6, 4), find Ker f.

iii) What can you generalize about Ker f from (i) and (ii)?

Answer 1

Let $f: \Z\to \Z_m \times \Z_n,\ f (x)=(x\mod m,\ x \mod n),\ \ m,n\in\N.$

i) If $(m, n) = (3, 4)$, then $\ker f=\{z\in\Z\ :\ f(z)=(0,0)\}=\{z\in\Z\ :\ (z\mod 3,\ z \mod 4)=(0,0)\}=\{z\in\Z\ :\ z\mod 3=0,\ z \mod 4=0\} =\{z\in\Z\ :\ 3|z,\ 4|z \}=\{z\in\Z\ :\ 12|z \}=12\Z.$

ii) If $(m, n) = (6, 4)$, then $\ker f=\{z\in\Z\ :\ f(z)=(0,0)\}=\{z\in\Z\ :\ (z\mod 6,\ z \mod 4)=(0,0)\}=\{z\in\Z\ :\ z\mod 6=0,\ z \mod 4=0\} =\{z\in\Z\ :\ 6|z,\ 4|z \}=\{z\in\Z\ :\ lcm(6,4)|z \}=\{z\in\Z\ :\ 12|z \}=12\Z.$

iii) We see that $\ker f$ from (i) and (ii) coincide. We can generalize that for two pairs $(m_1,n_1)$ and $(m_2,n_2)$ the kernels of corresponding maps coincide if and only if $lcm(m_1,n_1)=lcm(m_2,n_2).$