Answered

Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Ask anything and receive comprehensive, well-informed responses from our dedicated team of experts.

The prime numbers p, q, r satisfy the simultaneous equations
pq + pr = 80 and pq + qr = 425. Find the value of p + q + r.

Sagot :

Icuk18sugiartoidn
[tex]first\ equation :\\pq+pr=80\\pq=80-pr..........[1]\\\\secon\ equation :\\pq+qr=425..........[2][/tex]

[tex]Now\ substituting\ equation\ 1\ into\ 2:\\\\pq+qr=425\\(80-pr)+qr=425\\-pr+qr=425-80\\-pr+qr=345\\r(q-p)=345\\\\we\ have :\\ p(q+r)=80=2^{4} \times 5,\\q(p+r)=425=5^{2} \times 17\\r(q-p)=345=3 \times 5 \times 23\\\\so\ : p = 2\ or\ 5, ~q=5\ or\ 17,~ r = 3,5,or\ 23.[/tex]

[tex]From\ : p(q+r)=80, if~p=5~then\ q+r=16, which\ has\ no\ solution\\So~~p=2, then\ ~q+r=40, hence~~q=17~~and~~r=23.\\\\\therefore~~ p+q+r~~~is~~~\boxed{2+17+23=42}[/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.