Question-172662

Question Number 172662 by SANOGO last updated on 29/Jun/22

Answered by Mathspace last updated on 29/Jun/22

a) Δ(x,y)=((xy)/(M(x,y)))=((1512)/(252))=6 x=pd and y=qd with(p,d)=1 x+y=1512 ⇒d(p+q)=1512 ⇒p+q=((1512)/6)=252 ⇒(p,q)∈{(1,251),(2,250),(3,249)...(251,1)} we extract the couples wich verify Δ(p,q)=1... ⇒x=6p and y=6q...

$$\left.{a}\right)\:\Delta\left({x},{y}\right)=\frac{{xy}}{{M}\left({x},{y}\right)}=\frac{\mathrm{1512}}{\mathrm{252}}=\mathrm{6} \\ $$$${x}={pd}\:{and}\:{y}={qd}\:\:{with}\left({p},{d}\right)=\mathrm{1} \\ $$$${x}+{y}=\mathrm{1512}\:\Rightarrow{d}\left({p}+{q}\right)=\mathrm{1512} \\ $$$$\Rightarrow{p}+{q}=\frac{\mathrm{1512}}{\mathrm{6}}=\mathrm{252} \\ $$$$\Rightarrow\left({p},{q}\right)\in\left\{\left(\mathrm{1},\mathrm{251}\right),\left(\mathrm{2},\mathrm{250}\right),\left(\mathrm{3},\mathrm{249}\right)…\left(\mathrm{251},\mathrm{1}\right)\right\} \\ $$$${we}\:{extract}\:{the}\:{couples}\:{wich}\:{verify} \\ $$$$\Delta\left({p},{q}\right)=\mathrm{1}… \\ $$$$\Rightarrow{x}=\mathrm{6}{p}\:{and}\:{y}=\mathrm{6}{q}… \\ $$

Commented by SANOGO last updated on 30/Jun/22

$${thank}\:{you} \\ $$

Commented by Mathspace last updated on 30/Jun/22

$${x}+{y}={pd}+{qd}=\left({p}+{q}\right){d} \\ $$$$=\mathrm{6}.\mathrm{252}=\mathrm{1512} \\ $$

Answered by puissant last updated on 30/Jun/22

b) posons δ=pgcd(x;y) dire cela renvoi a: ∃ (x′;y′)∈N^2 / x=δx′ ; y=δy′ et x′∧y′=1 xy=pgcd(x;y)•ppcm(x;y) ⇒ ppcm(x;y)=δx′y′=1440 en meme temps, δ(x′+y′)=276 donc δ/1440 et δ/276.... A toi de conctinuer.

$$\left.{b}\right)\: \\ $$$${posons}\:\delta={pgcd}\left({x};{y}\right)\:{dire}\:{cela}\:{renvoi}\:{a}: \\ $$$$\exists\:\left({x}';{y}'\right)\in\mathbb{N}^{\mathrm{2}} /\:{x}=\delta{x}'\:;\:\:{y}=\delta{y}'\:{et}\:{x}'\wedge{y}'=\mathrm{1} \\ $$$${xy}={pgcd}\left({x};{y}\right)\bullet{ppcm}\left({x};{y}\right) \\ $$$$\Rightarrow\:{ppcm}\left({x};{y}\right)=\delta{x}'{y}'=\mathrm{1440} \\ $$$${en}\:{meme}\:{temps},\:\delta\left({x}'+{y}'\right)=\mathrm{276} \\ $$$${donc}\:\delta/\mathrm{1440}\:{et}\:\delta/\mathrm{276}…. \\ $$$$ \\ $$$${A}\:{toi}\:{de}\:{conctinuer}. \\ $$

Answered by mr W last updated on 30/Jun/22

a) xy=1512=2^3 ×3^3 ×7 LCM=252=2^2 ×3^2 ×7 number 1= 2×3×1 number 2= 2^2 ×3^2 ×7 (x, y)=(2×3×1, 2^2 ×3^2 ×7)=(6, 252) or (252, 6) (x, y)=(2^2 ×3×1, 2×3^2 ×7)=(12, 126) or (126, 12) (x, y)=(2×3^2 ×1, 2^2 ×3×7)=(18, 84) or (84, 18) (x, y)=(2×3×7, 2^2 ×3^2 ×1)=(42, 36) or (36, 42)

$$\left.{a}\right) \\ $$$${xy}=\mathrm{1512}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{7} \\ $$$${LCM}=\mathrm{252}=\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{7} \\ $$$${number}\:\mathrm{1}=\:\mathrm{2}×\mathrm{3}×\mathrm{1} \\ $$$${number}\:\mathrm{2}=\:\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{7} \\ $$$$\left({x},\:{y}\right)=\left(\mathrm{2}×\mathrm{3}×\mathrm{1},\:\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{7}\right)=\left(\mathrm{6},\:\mathrm{252}\right)\:{or}\:\left(\mathrm{252},\:\mathrm{6}\right) \\ $$$$\left({x},\:{y}\right)=\left(\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{1},\:\mathrm{2}×\mathrm{3}^{\mathrm{2}} ×\mathrm{7}\right)=\left(\mathrm{12},\:\mathrm{126}\right)\:{or}\:\left(\mathrm{126},\:\mathrm{12}\right) \\ $$$$\left({x},\:{y}\right)=\left(\mathrm{2}×\mathrm{3}^{\mathrm{2}} ×\mathrm{1},\:\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{7}\right)=\left(\mathrm{18},\:\mathrm{84}\right)\:{or}\:\left(\mathrm{84},\:\mathrm{18}\right) \\ $$$$\left({x},\:{y}\right)=\left(\mathrm{2}×\mathrm{3}×\mathrm{7},\:\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{1}\right)=\left(\mathrm{42},\:\mathrm{36}\right)\:{or}\:\left(\mathrm{36},\:\mathrm{42}\right) \\ $$

Answered by Rasheed.Sindhi last updated on 30/Jun/22

(c) Let x=24m & y=24n where (m,n)=1 xy=24m∙24n=576mn=16128 mn=28 mn=1∙28=28∙1 mn=4∙7=7∙4 (m,n)=(1,28),(28,1),(4,7),(7,4) (x,y)=(24m,24n) =(24,672),(672,24),(96,168),(168,96)

$$\left(\boldsymbol{\mathrm{c}}\right)\:\:\:{Let}\:{x}=\mathrm{24}{m}\:\&\:{y}=\mathrm{24}{n}\:{where}\: \\ $$$$\left({m},{n}\right)=\mathrm{1} \\ $$$${xy}=\mathrm{24}{m}\centerdot\mathrm{24}{n}=\mathrm{576}{mn}=\mathrm{16128} \\ $$$${mn}=\mathrm{28} \\ $$$${mn}=\mathrm{1}\centerdot\mathrm{28}=\mathrm{28}\centerdot\mathrm{1} \\ $$$${mn}=\mathrm{4}\centerdot\mathrm{7}=\mathrm{7}\centerdot\mathrm{4} \\ $$$$\left({m},{n}\right)=\left(\mathrm{1},\mathrm{28}\right),\left(\mathrm{28},\mathrm{1}\right),\left(\mathrm{4},\mathrm{7}\right),\left(\mathrm{7},\mathrm{4}\right) \\ $$$$\left({x},{y}\right)=\left(\mathrm{24}{m},\mathrm{24}{n}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{24},\mathrm{672}\right),\left(\mathrm{672},\mathrm{24}\right),\left(\mathrm{96},\mathrm{168}\right),\left(\mathrm{168},\mathrm{96}\right) \\ $$

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