If a and b are positive integers and 90a = b^3, which of the following

Question

If a and b are positive integers and  90a = b^3 , which of the following must be an integer?

I.  \frac{a}{2^3*3*5}

II.  \frac{a}{2*3^2*5}

III.  \frac{a}{2^2*3 *5^2}

(A) Only I
(B) Only II
(C) Only III
(D) Only I and II
(E) I, II and III

chetan2u · Accepted Answer

Now  90a=b^3.......2*3^2*5*a=b^3 
So a has to be a multiple of 2^2*3*5^2...
Thus a divided by a factor of 2^2*3*5^2 will be an integer..

I.  \frac{a}{2^2*3*5} 
2^2*3*5 is a factor of 2^2*3*5^2....so YES

II.  \frac{a}{2*3^2*5} 
2*3^2*5 is NOT a factor of 2^2*3*5^2....so Not necessary

III.  \frac{a}{2^2*3 *5^2} 
2^2*3*5^2 is a factor of 2^2*3*5^2....so YES

I and III

Bunuel , no choice matches the answer, so there must be a typo in choice I

Bunuel · Answer

You are right. Edited option I. Thank you.

KSBGC · Answer

90a = b^3 . So, 90a is a perfect cube.

90 = 3*3*5*2*a.

a = 3*5^2*2^2  = 300

scan each option:

I.  \frac{a}{2^3*3*5}  = 300 / 120 = no integer.

II.  \frac{a}{2*3^2*5}  = 300 / 90 = no integer.

III.  \frac{a}{2^2*3 *5^2}  = 300/ 300 = integer .

C is the best answer.

mskx · Answer

Don't get the step highlighted in red. Could someone be so kind and explain it to me?

KSBGC · Answer

90a = b^3

90a has to be a perfect cube. How can we make a perfect cube?

If we break down 90 what we get:

90 = 3 * 3*5 * 2 .

how many 3 u have and what's about 5 and 2. u have 3^2 and a 5 and a 2. Still we don't know about a.

when u say something is a perfect cube it means all the prime factors of that number must have to be expressed through cube (3)

3^2*5*2 is not perfect cube. So rest of the elements come from a. remember that 90a is perfect cube. but 90 alone is not a perfect cube.

3^2 ..........need another 3

2 .....need  2^2

5.......need  5^2

Therefore this additional properties is the part of a.

a = 3 * 2^2 * 5^2 = 300

now see:  90a = 90*300 = b^3

shuvodip04 · Answer

Since 90a=b^3 -> 3*3*2*5a=b^3
Since b is a positive integer then LHS of above has to be a perfect cube.

so a=2*5*2*5*3=300 ,Now if we put the value of a=300 in each of the options I,II and III, only III returns a integral value.

ScottTargetTestPrep · Answer

We see that 90a must be a perfect cube. Thus 90a must have prime factors whose exponents are multiples of 3. Let’s break 90 into prime factors:

90 = 9 x 10 = 2^1 x 3^2 x 5^1

So in order for 90a to be a perfect cube, the number “a” must contain the product 2^2 x 3^1 x 5^2.

Thus, only III must be an integer.

Answer: C

bumpbot · Answer

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If a and b are positive integers and 90a = b^3, which of the following

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If a and b are positive integers and 90a = b^3, which of the following

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