# n-N-b-a-N-a-0-In-base-10-n-aabb-1-show-that-n-is-not-prime-2-Give-conditions-on-b-such-that-n-is-perfect-square-3-Determinate-n-such-that-n-is-a-perfect-square-

Question Number 142388 by mathocean1 last updated on 30/May/21
$$\mathrm{n}\:\in\:\mathbb{N},\:\mathrm{b},\:\mathrm{a}\:\in\:\mathbb{N}\:;\:\mathrm{a}\neq\mathrm{0}. \\$$$$\mathrm{In}\:\mathrm{base}\:\mathrm{10};\:\mathrm{n}=\overline {\mathrm{aabb}}\: \\$$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{prime}. \\$$$$\mathrm{2}.\:\mathrm{Give}\:\mathrm{conditions}\:\mathrm{on}\:\mathrm{b}\:\mathrm{such}\:\mathrm{that} \\$$$$\mathrm{n}\:\mathrm{is}\:\mathrm{perfect}\:\mathrm{square}. \\$$$$\mathrm{3}.\:\mathrm{Determinate}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\: \\$$$$\mathrm{perfect}\:\mathrm{square}. \\$$
Answered by mr W last updated on 31/May/21
$$\mathrm{1}. \\$$$${n}=\mathrm{1000}{a}+\mathrm{100}{a}+\mathrm{10}{b}+{b} \\$$$$=\mathrm{1100}{a}+\mathrm{11}{b} \\$$$$=\mathrm{11}\left(\mathrm{100}{a}+{b}\right)\equiv\mathrm{0}\left({mod}\:\mathrm{11}\right) \\$$$$\mathrm{2}. \\$$$$\mathrm{100}{a}+{b}=\mathrm{11}{m}^{\mathrm{2}} \geqslant\mathrm{101}\:\Rightarrow{m}\geqslant\mathrm{4} \\$$$$\mathrm{100}{a}+{b}=\mathrm{11}{m}^{\mathrm{2}} \leqslant\mathrm{909}\:\Rightarrow{m}\leqslant\mathrm{9} \\$$$${b}=\mathrm{11}{m}^{\mathrm{2}} −\mathrm{100}{a} \\$$$$\mathrm{3}. \\$$$${n}=\mathrm{11}^{\mathrm{2}} {m}^{\mathrm{2}} =\left(\mathrm{11}{m}\right)^{\mathrm{2}} \:{with}\:\mathrm{4}\leqslant{m}\leqslant\mathrm{9} \\$$
Commented by mathocean1 last updated on 31/May/21
$$\mathscr{S}{orry}…{please}\:{sir};\:{according}\:{to}\:{the}\: \\$$$${second}\:{question}\:{why}\:{have}\:{you}\: \\$$$$\mathrm{100}{a}+{b}\leqslant\mathrm{909}\:{and}\:\geqslant\mathrm{101}\:? \\$$
Commented by mr W last updated on 31/May/21
$$\mathrm{1}\leqslant{a},{b}\leqslant\mathrm{9} \\$$$$\mathrm{100}{a}+{b}\leqslant\mathrm{100}×\mathrm{9}+\mathrm{9}=\mathrm{909} \\$$$$\mathrm{100}{a}+{b}\geqslant\mathrm{100}×\mathrm{1}+\mathrm{1}=\mathrm{101} \\$$
Commented by mathocean1 last updated on 31/May/21
$${thanks} \\$$