Question Number 125672 by mathocean1 last updated on 12/Dec/20
![N=x32y in base 10. N≡0[3] and N≡0[4]. N>8329 N has four digits. Determinate N.](Q125672.png)
$${N}={x}\mathrm{32}{y}\:{in}\:{base}\:\mathrm{10}.\: \\ $$ $${N}\equiv\mathrm{0}\left[\mathrm{3}\right]\:{and}\:{N}\equiv\mathrm{0}\left[\mathrm{4}\right]. \\ $$ $${N}>\mathrm{8329} \\ $$ $${N}\:{has}\:{four}\:{digits}. \\ $$ $${Determinate}\:{N}. \\ $$
Answered by JDamian last updated on 12/Dec/20
![x+3+2+y≡0[3] → x+y≡1[3] N>8329 → x=9 → y≡1[3] → y∈{1, 4, 7} N≡0[4] → 2×2+y≡0[4] → y≡0[4] y=4 N=9324](Q125679.png)
$${x}+\mathrm{3}+\mathrm{2}+{y}\equiv\mathrm{0}\left[\mathrm{3}\right]\:\:\rightarrow\:\:{x}+{y}\equiv\mathrm{1}\left[\mathrm{3}\right] \\ $$ $${N}>\mathrm{8329}\:\:\rightarrow\:\:\boldsymbol{{x}}=\mathrm{9}\:\:\:\rightarrow\:\:{y}\equiv\mathrm{1}\left[\mathrm{3}\right]\:\:\rightarrow\:\:{y}\in\left\{\mathrm{1},\:\mathrm{4},\:\mathrm{7}\right\} \\ $$ $$ \\ $$ $${N}\equiv\mathrm{0}\left[\mathrm{4}\right]\:\:\rightarrow\:\:\mathrm{2}×\mathrm{2}+{y}\equiv\mathrm{0}\left[\mathrm{4}\right]\:\:\rightarrow\:\:{y}\equiv\mathrm{0}\left[\mathrm{4}\right] \\ $$ $$\boldsymbol{{y}}=\mathrm{4} \\ $$ $$ \\ $$ $$\boldsymbol{{N}}=\mathrm{9324} \\ $$