Question

What is the smallest square number which is sivisible by 24569?

Answer 1

Answer:

it's 900

Step-by-step explanation:

we multiply the LCM 90 by 2*5, hence required smallest square number is 900.

Answer 2

The given question is What is the smallest square number which is divisible by 2,4,5,6,9?

we have to find the smallest number which is divisible by all the given numbers.

The given numbers are 2,4,5,6,9.

The solution to this question is obtained by taking the LCM of the given numbers.

The LCM for the given numbers is obtained by a prime factorisation method.

The prime factorisation of 2,4,5,6,9 is

$2)2 \: 4 \: 5 \: 6 \: 9 \\ 2)1 \: 2 \: 5 \: 3 \: 9 \\2 )1 \: 1 \: 5 \: 3 \: 9 \\ 3)1 \: 1 \: 5 \: 1 \: 3 \\ 3)1 \: 1 \: 5 \: 1 \: 1\\ 5)1 \: 1 \: 1 \: 1 \: 1$

The LCM of 2,4,5,6,9, is

${2}^{2} \times {3}^{2} \times {5}^{1}$

The LCM of the given numbers will be 180.

But we need the smallest square number, where the LCM of these numbers is not a square.

On seeing the factors it is found that 2 and 3 appear 2 times which is a square but only 5 is a single number.

so multiplying by the number 5 we get a square number

${2}^{2} \times {3}^{2} \times {5}^{2} = 180 \times 5 = 900$

where 900 is a square number.

Therefore, the smallest square number which is divisible by 24569 is 900.

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