A-number-n-leaves-a-remainder-of-22-when-divided-by-24-and-remainder-30-when-divided-by-33-Find-the-least-possible-value-of-n-

Question Number 86141 by TawaTawa1 last updated on 27/Mar/20

$$\mathrm{A}\:\mathrm{number}\:\mathrm{n}\:\mathrm{leaves}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{of}\:\:\mathrm{22}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{24}\:\mathrm{and} \\ $$$$\mathrm{remainder}\:\:\mathrm{30}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{33}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{n} \\ $$

Commented by mr W last updated on 27/Mar/20

$${there}\:{is}\:{no}\:{such}\:{number}! \\ $$

Commented by Kunal12588 last updated on 27/Mar/20

n=24p+22 n=33q+30 24p−33q=8 ⇒24p−33p+33δ=0 [let q=p−δ] ⇒33δ−9p=8 ⇒11δ−3p=(8/3) ⇒ no solution in natural no.

$${n}=\mathrm{24}{p}+\mathrm{22} \\ $$$${n}=\mathrm{33}{q}+\mathrm{30} \\ $$$$\mathrm{24}{p}−\mathrm{33}{q}=\mathrm{8} \\ $$$$\Rightarrow\mathrm{24}{p}−\mathrm{33}{p}+\mathrm{33}\delta=\mathrm{0}\:\:\:\left[{let}\:{q}={p}−\delta\right] \\ $$$$\Rightarrow\mathrm{33}\delta−\mathrm{9}{p}=\mathrm{8} \\ $$$$\Rightarrow\mathrm{11}\delta−\mathrm{3}{p}=\frac{\mathrm{8}}{\mathrm{3}} \\ $$$$\Rightarrow\:{no}\:{solution}\:{in}\:{natural}\:{no}. \\ $$

Commented by Kunal12588 last updated on 27/Mar/20