Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 110592 by Aina Samuel Temidayo last updated on 29/Aug/20

How many pairs of integers x and y satisfy the equation (1/x)+(1/y)=(1/(32))

$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{32}} \\ $$

Answered by Aziztisffola last updated on 29/Aug/20

if x=y ⇒ (2/x)=(1/(32))⇒x=64 (x;y)=(64;64) ∈S then S≠∅ there is at least one solution.

$$\:\mathrm{if}\:\mathrm{x}=\mathrm{y}\:\Rightarrow\:\frac{\mathrm{2}}{\mathrm{x}}=\frac{\mathrm{1}}{\mathrm{32}}\Rightarrow\mathrm{x}=\mathrm{64} \\ $$$$\:\left(\mathrm{x};\mathrm{y}\right)=\left(\mathrm{64};\mathrm{64}\right)\:\in\mathrm{S}\:\mathrm{then}\:\mathrm{S}\neq\varnothing \\ $$$$\mathrm{there}\:\mathrm{is}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{solution}.\: \\ $$

Answered by mr W last updated on 29/Aug/20

(1/y)=(1/(32))−(1/x)=((x−32)/(32x)) y=((32x)/(x−32)) x=33 ⇒y=1056 x=34 ⇒y=544 x=36 ⇒y=288 x=40 ⇒y=160 x=48 ⇒y=96 x=64 ⇒y=64 x=31 ⇒y=−993 x=30 ⇒y=−480 x=28 ⇒y=−224 x=24 ⇒y=−96 x=16 ⇒y=−32 ......

$$\frac{\mathrm{1}}{{y}}=\frac{\mathrm{1}}{\mathrm{32}}−\frac{\mathrm{1}}{{x}}=\frac{{x}−\mathrm{32}}{\mathrm{32}{x}} \\ $$$${y}=\frac{\mathrm{32}{x}}{{x}−\mathrm{32}} \\ $$$${x}=\mathrm{33}\:\Rightarrow{y}=\mathrm{1056} \\ $$$${x}=\mathrm{34}\:\Rightarrow{y}=\mathrm{544} \\ $$$${x}=\mathrm{36}\:\Rightarrow{y}=\mathrm{288} \\ $$$${x}=\mathrm{40}\:\Rightarrow{y}=\mathrm{160} \\ $$$${x}=\mathrm{48}\:\Rightarrow{y}=\mathrm{96} \\ $$$${x}=\mathrm{64}\:\Rightarrow{y}=\mathrm{64} \\ $$$${x}=\mathrm{31}\:\Rightarrow{y}=−\mathrm{993} \\ $$$${x}=\mathrm{30}\:\Rightarrow{y}=−\mathrm{480} \\ $$$${x}=\mathrm{28}\:\Rightarrow{y}=−\mathrm{224} \\ $$$${x}=\mathrm{24}\:\Rightarrow{y}=−\mathrm{96} \\ $$$${x}=\mathrm{16}\:\Rightarrow{y}=−\mathrm{32} \\ $$$$...... \\ $$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

Thanks but how did you know the values of (x,y) that fits the equation? Your work is not well explanatory.

$$\mathrm{Thanks}\:\mathrm{but}\:\mathrm{how}\:\mathrm{did}\:\mathrm{you}\:\mathrm{know}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{that}\:\mathrm{fits}\:\mathrm{the}\:\mathrm{equation}?\:\mathrm{Your}\:\mathrm{work}\:\mathrm{is} \\ $$$$\mathrm{not}\:\mathrm{well}\:\mathrm{explanatory}. \\ $$

Commented by mr W last updated on 29/Aug/20

if x−32 divides 32, then x is ok. ⇒x−32=±1,±2,±4,±8,±16,±32

$${if}\:{x}−\mathrm{32}\:{divides}\:\mathrm{32},\:{then}\:{x}\:{is}\:{ok}. \\ $$$$\Rightarrow{x}−\mathrm{32}=\pm\mathrm{1},\pm\mathrm{2},\pm\mathrm{4},\pm\mathrm{8},\pm\mathrm{16},\pm\mathrm{32} \\ $$

Answered by floor(10²Eta[1]) last updated on 29/Aug/20

go see my answer on Q.103093 it′s the same logic

$$\mathrm{go}\:\mathrm{see}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{on}\:\mathrm{Q}.\mathrm{103093} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{same}\:\mathrm{logic} \\ $$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

Thanks.

$$\mathrm{Thanks}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com