Solving for (ab,ac,bc) = 1: Where Am I Going Wrong?

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The discussion revolves around the equation (ab, ac, bc) = 1 and seeks clarification on the notation and implications of this expression. Participants express confusion about the meaning of "a, b, c =" and whether it indicates that all three variables equal the right side of the equation. The relationship between the greatest common divisor (gcd) and least common multiple (lcm) is highlighted, with questions about how they relate to the given equation. There is a consensus that if a prime p divides one of the products, it must divide the others, but the reasoning behind the initial equation remains unclear. The thread ultimately seeks guidance on understanding the notation and proving the stated relationship.
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i have this one ? and it is bugging me ! show that a,b,c = to (ab,ac,bc)[a,b,c]. () = to the gcd and [] = to the lcm. does that notation mean multiply the numbers together? i mean i started out saying this...

(ab,ac,bc) = 1, so there exist a p prime s.t. p divides ab, p divides ac, and p divides bc. so if p divides ac, then p HAS to divide the other two, right? can anyone show me where i am going wrong and point me in the right direction? thank you all!
 
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1+1=1 said:
i have this one ? and it is bugging me ! show that a,b,c = to (ab,ac,bc)[a,b,c].
What does that mean, specifically, "a,b,c = "? Do you mean all three numbers, a, b, and c equal the right side?

(ab,ac,bc) = 1
Why would this be true?
 
Okay, (a,b,c) is the gdc and [abc] is the lcm but what does a,b,c on the left side of the equation mean? I thought at first that you had forgotten the "[" and meant [a,b,c] but then it's not true. Do you mean a*b*b= (a,b,c)*[a,b,c]?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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