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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.

$${N}={x}\mathrm{32}{y}\:{in}\:{base}\:\mathrm{10}.\: \\ $$ $${N}\equiv\mathrm{0}\left[\mathrm{3}\right]\:{and}\:{N}\equiv\mathrm{0}\left[\mathrm{4}\right]. \\ $$ $${N}>\mathrm{8329} \\ $$ $${N}\:{has}\:{four}\:{digits}. \\ $$ $${Determinate}\:{N}. \\ $$

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

$${x}+\mathrm{3}+\mathrm{2}+{y}\equiv\mathrm{0}\left[\mathrm{3}\right]\:\:\rightarrow\:\:{x}+{y}\equiv\mathrm{1}\left[\mathrm{3}\right] \\ $$ $${N}>\mathrm{8329}\:\:\rightarrow\:\:\boldsymbol{{x}}=\mathrm{9}\:\:\:\rightarrow\:\:{y}\equiv\mathrm{1}\left[\mathrm{3}\right]\:\:\rightarrow\:\:{y}\in\left\{\mathrm{1},\:\mathrm{4},\:\mathrm{7}\right\} \\ $$ $$ \\ $$ $${N}\equiv\mathrm{0}\left[\mathrm{4}\right]\:\:\rightarrow\:\:\mathrm{2}×\mathrm{2}+{y}\equiv\mathrm{0}\left[\mathrm{4}\right]\:\:\rightarrow\:\:{y}\equiv\mathrm{0}\left[\mathrm{4}\right] \\ $$ $$\boldsymbol{{y}}=\mathrm{4} \\ $$ $$ \\ $$ $$\boldsymbol{{N}}=\mathrm{9324} \\ $$

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