Find the least squared number exactly divisible by 12,16,25 and 40.

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mathewww's profile pic

mathewww | High School Teacher | (Level 1) Honors

Posted on

The least number that is exactly divisible by 12, 16, 25 and 40 is the least common multiple (LCM) of these numbers.

By division method, the LCM of these numbers work out to be 3*2*2*2*5*2*5= 1200. 

So, in order to find out the least squared number that is exactly divisible by 12, 16, 25 and 40, we have to find out the least multiple of 1200 which is a perfect square.

Inspection of the prime factors of 1200 reveals we have a prime factor 3 which has no pair, so, in order to make the number 1200 a perfect square, we need to multiply it by 3.  The desired number is thus 1200*3 = 3600.

Therefore, the least number that is exactly divisible by 12, 16, 25 and 40 is 3600.

pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

Let  number be  `x^2`  (which is least) is divisible by 12,16,25 and 40.
Since x^2 is divisible by 12, it is divisible by 3 and 4.
But then 3 divides x, so x^2 is divisible by 9.
If 4 divides `x^2` , then it is not necessarily true that 4 divides x. However,if  16 divides `x^2` .
If 25  divides  `x^2` .So
If `x^2=9xx16xx25`  
Then we know that 40 will  divide `x^2=9xx2xx8xx5xx5`
Thus, we simply take `x^2=9xx16xx25=3600=60^2`.
This is the least such number.

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