# Find the smallest value of k, when, a) 280k is a perfect square, b) 882k is a cube.

neela | High School Teacher | (Level 3) Valedictorian

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To find k such that 280k is a suare and 882k is a perfect cube.

280k = 8*35k =  2^3*5*7k

280k = 2^3*5*7*k

882k = 2*441k = 2*21^2*k = 2*3^2*7^2*k.

882k = 2*3^2*7^2k

Therefore we write 280 and 882 as powers of their prime factors as below:

280k = (2^3)(3^0)(5^1)(7^1)k

882k = (2^1)(3^2)(5^0)(7^2)k.

Now consider 2^3 and 2^1 :  If we multiply 2^5 both 2^3 becomes a square of(2^3) and 2^1*2^5 = (2^2)^3.

Now consider 3^0 and 3^2: multiplying by 3^4 makes 3^0*3^4 = (3^2)^2 and (3^2*3^4)  =  (3^2)^3.

Now consider 5^1 and 5^0: 5^3 is the multiplier, which makes (5^1*5^3) = (5^2)^2 and  5^0*5^3 = (5)^3.

Now consider 7^1 and 7^2 : The multiplier is 7^1, which makes 7^1*7^1 = 7^2 and 7^2*7^1 = (7^3).

Therefore the required least multiplier is k = ( 2^5)(3^4)(5^3)(7^1) = 32*81*1125*7 = 2268000.

Tally:

280* 2268000 = (25200)^2

882*2268000 = (1260)^3

sociality | High School Teacher | (Level 1) Valedictorian

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Here we have to find the smallest value of k for which 280k is a perfect square and 882k is a perfect cube.

Now, writing 280 as a product of prime numbers, we get 7*2*2*2*5. Writing 882 as a product of prime numbers we get 3*3*2*7*7.

Now, for 280k to be a perfect square all the prime factors should be in pairs, and if 882k has to be a perfect cube all the prime factors should be in triplets. Or in other words 280k should have a square of every relevant prime number and 882k should have a cube of the same prime numbers.

For this, we need k to have one factor equal to 32 as 32*8 is a square and 32*2 is a cube. Also k needs to have another factor of 7 as 7*7 is a square and 7*49 is a cube. Similarly, k needs to have a factor of 125 as 125*5 is a square and 125*1 is a cube. And finally k needs to have a factor of 81 as 81*1 is a square and 81*9 is a cube.

In this way we arrive at the factors of k as 32, 81, 125 and 7. So k is equal to 2268000.

We see that 280*2268000 is a perfect square and 882*2268000 is a perfect cube. This is not possible with any value of k smaller than 2268000.