The numbers 225 and 540, written as the products of their prime factors,are 225=3^2 x 5^2 , 540=2^2 x 3^3 x 5?

Question

(a) Write 2250 as the product of it's prime factors.
(b) Find the smallest positive integer value of n for which 540n is a perfect square.

Answer 1

a. You're asked to do prime factorization. Check out my Math Variety blog posts on prime factorization for an explanation of this concept, practice problems, and solutions. Click on ads/links of interest:

http://mathvariety.blogspot.com/2009/04/prime-factorization.html

http://mathvariety.blogspot.com/2009/05/prime-factorization-practice-problems.html

http://mathvariety.blogspot.com/2009/05/prime-factorization-solutions-to.html



I also have an eHow article titled “How to Do Prime Factorization” at http://www.ehow.com/how_4776386_do-prime-factorization.html if you’re interested.



2,250 is:

2*(1,125)

2*(5*225)

2*5*(5*45)

2*5*5*(45)

2*5*5*5*(9)

2*5*5*5*(3*3)

2*3*3*5*5*5 or 2*3^2*5^3 (answer)



A shorter way would be to use the fact you're given 225 is 3^2*5^2. Now, 2,250 is 3^2*5^2*(5*2) which is just 2*3^2*5^3 which is the same answer.\



b. For the second one, list the multiples of 540:

540,1080,1620,2160,2700,3240,3780,4320,4860,5400, 5940,6480,7020,7560,8100,...



8100 is a perfect square since sqrt(8,100)=90. 540n=8,100 implies n=15. 15 is the answer.

Answer 2

(a)

2250

= 225 x 10

= 3² x 5² x 10

= 3² x 5² x 2 x 5

= 2 x 3² x 5³



(b)

540

= 3 x 3 x 5 x 3 x 2 x 2

= (3 x 3) x (2 x 2) x (3) x (5)



the lone pair is 3 and 5

multiply 3 and 5: 15

so smallest value of n is 15



double check:

540 x 15 = 8100

√8100 = 90

Answer 3

let f(x) = x/(x-3) find points where f(x) = -4: x = -4(x-3) x = -4x + 12 5x = 12 x = 12/5, so: f(12/5) = -4 We need to know where in the x-range this function produces higher/lower values. For this we take the derivative of f(x) with the quotation rule: f'(x) = (1.(x-3) - x.1) / (x-3)² = -3 / (x-3)² this derivative is always negative, except for x=3 where it is undefined. this means that f(x) will decrease for increasing x as long as x doesn't cross the value 3. so this means that f(x) < -4 for x in the interval (12/5, 3) for x > 3 itis clear that f(x) is always positive. conclusion: x < 12/5 or x > 3 nb: The first (edit: andsecond) poster made an error by multiplying with (x-3) which may change the inequality (<) when x-3 is negative. edit: I see second poster also made an error. please test these solutions: for example: x = -10 => the condition holds x/(x-3) > -4 x = 13/5 => the condition does not hold x/(x-3) < -4 so x = -10 shoud be inside the range and x = 13/5 not.

Answer 4

(a) 2250=2x3^2x5^3



(b) 540n=perfect square

n=3x5=15

so 540n=540x15=8100=2^2x3^4x5^2

its square root is 2x3^2x5=90