The least positive integer that is divisible by 2,3,4 and 5, and is

Question

The least positive integer that is divisible by 2, 3, 4 and 5, and is also a perfect square, perfect cube, 4th power and 5th power, can be written in the form \(a^b\) for positive integers a and b. What is the least positive value of \(a+b\) ?

(A) 80
(B) 85
(C) 90
(D) 95
(E) 100

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Official Answer

C

(A) 80
(B) 85
(C) 90
(D) 95
(E) 100

Are You Up For the Challenge: 700 Level Questions

Kinshook · Answer

Given: The least positive integer that is divisible by 2, 3, 4 and 5, and is also a perfect square, perfect cube, 4th power and 5th power, can be written in the form  a^b  for positive integers a and b.

Asked: What is the least positive value of  a+b  ?

Least positive integer divisible by 2,3,4 & 5 = LCM(2,3,4,5) = 60

Let us consider aˆb = 30ˆ60.
It is divisible by 2,3,4 & 5 and is also a perfect square, perfect cube, 4th power and 5th power, can be written in the form  a^b  for positive integers a and b.

a + b = 30 + 60 = 90

IMO C

bv8562 · Answer

Kinshook  How is  60^{30}  a 4th power?

Kinshook · Answer

It is a 4th power of 2

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IanStewart · Answer

It's not a 4th power:

60^{30} = (2^2 	imes 3 	imes 5)^{30} = 2^{60} 3^{30} 5^{30}

If this number were a 4th power, every exponent in its prime factorization would need to be a multiple of 4, but that's not the case. The number is a square, a cube, and a 5th power, because the prime factorization exponents are multiples of 2, 3 and 5, so it's close to meeting the requirements but isn't quite right.

The smallest number that works here is  30^{60} . That number is clearly a power of 2, 3, 4 and 5, because the exponent is a multiple of 2, 3, 4 and 5. It's also clearly divisible by 2, 3 and 5 because the base 30 is divisible by those numbers, but it will also certainly be a multiple of 4 = 2^2, because we'll end up finding 2^60 in its prime factorization. So the answer is 30 + 60 = 90, since there's no smaller possibility.

bv8562 · Answer

Thanks  IanStewart  for clarifying my doubt and this is exactly what I was going to ask to  Kinshook

Its a perfect square because:  (60^{15})^2 
Its a cube because:  (60^{10})^3 
Its a 5th power because:  (60^{6})^5

But it is not a 4th power because 30 is not a multiple of 4. However, if we interchange the values of the base and the exponent then that makes sense i.e  30^{60}  as it satisfies all the given conditions.

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The least positive integer that is divisible by 2,3,4 and 5, and is

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