Can't figure out how the answer was arrived at :(

I have a couple questions that I know the solutions to, but I can't for the life of me figure out the process involved at arriving at those solutions...

1) If 18*sqrt(18) = r*sqrt(t), where t and r are positive integers and r > t, which of the following could be the value of r*t? (solution is 108)

2) The eggs in a certain basket are either white or brown. If the ration of the number of white egges to the number of brown eggs is (2/3), each of the following could be the number of eggs in the basket EXCEPT:

a) 10

b) 12 (this is the answer)

c) 15

d) 30

e) 60

I thought about proportionalities with problem 1 and got no where, and with problem 2 I thought I had it nailed until I checked the answer.

As always, any advice is greatly appreciated :)

Re: Can't figure out how the answer was arrived at :(

these are not university-level algebra questions.

for (1):

since everything in sight is positive we can square both sides without fear.

thus from:

18√18 = r√t, we have:

(324)(18) = r^{2}t

18^{3} = 5832 = r^{2}t.

clearly t < 18, or else r^{2}t > t^{3} > 5832. so t is some divisor of 18: 1,2,3,6,or 9.

if t = 1, r = √(5832), which is not an integer.

if t = 2, r = √(2916) = 54 <--this works ( (18)^{3}/2 = (9)(18)^{2}, which has square root 3*18 = 54).

if t = 3, r = √(1944), not an integer

if t = 6, r = √(972), not an integer

if t = 9, r = √(648), not an integer

(look at the prime factorization of 18 cubed)

so the only case where r and t are integers with r > t is t = 2, r = 54, hence rt = 108.

(2) 2x + 3x = 5x, the number of eggs in the basket must be a multiple of 5. 12 is not a multiple of 5.

Re: Can't figure out how the answer was arrived at :(

Thanks Denovo, and I just realized I was posting these questions in the incorrect place (my apologies to all, I see where to post these now). Thanks again for your help.