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**The Problem:**

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 but I have no clue how this war arrived at)

**The Response:**

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.

**My follow up to the response:**

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

thus from:

thus from:

*√18 = r√t, we have:*

(324)(18) = r

18

clearly t < 18, or else r

(324)(18) = r

^{2}t18

^{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."I think where I'm getting hung up in the explanation is the "

*."*

**or else r**__clearly t < 18__,^{2}t > t^{3}> 5832.__1,2,3,6,or 9__**so t is some divisor of 18:**I understand that if t > r that the result would be > 5832. The part I'm having trouble with is how to indentify that t is in fact less than 18 specifically.How do we know this? And how does the responder know to use a factor/divisor of 18 and not some other integer?

Is there a link/website or some section of math one would practice in a book/class for a problem of this nature? It's important to me to understand the "why" of it instead of memorizing "how" to do a particular problem type.

Thanks.