find the smallest square number which is exactly divided by each of the number 12,24 and 40
Answers
Step-by-step explanation:
One thing you need to know about perfect squares: if you work out the prime factorization (hereafter, PF) of a perfect square, each of its factors appears an even number of times. For example 3*3*5 = 45. Since the factor 5 only appears once (and one is not even), we know that 45 isn’t a perfect square. But 3*3*5*5 = 225, which is a perfect square. We know it’s a perfect square because each of its factors appears twice, and 2 is even.
The PF of 10 is 2*5. 10 is not a perfect square, because 2 and 5 only appear once, and one is odd.
The PF of 16 is 2*2*2*2. 16 is a perfect square, because 2 appears four times, and four is even.
The PF of 24 is 2*2*2*3. 24 is not a perfect square, because 2 appears three times, and 3 appears once. Three and one are both odd numbers.
Any multiple of 10, 16, and 24 must contain each of their prime factors at least as many times as they appear in the numbers being multiplied. So for example, the LCM (least common multiple) of 10, 16, and 25 must contain four factors of 2 (since 2 appears four times in the PF of 16), one factor of 3 (since 3 appears once in the PF of 24, and nowhere else), and one factor of 5 (since 5 appears once in the PF of 10, and nowhere else. Here is the PF for the LCM of 10, 16, and 24:
2*2*2*2*3*5 = 240
But 240 isn’t a perfect square, since the factors 3 and 5 appear only once.
If we multiply 240 by any other number, the result will still be divisible by 10, 16, and 24. And since the number we’re seeking has even numbers of prime factors, we can simply multiply 240 by 3*5 (or 15) to get the smallest perfect square that is divisible by 10, 16, and 24:
2*2*2*2*3*3*5*5 = 3600.
There you have it: 3600 is the smallest square number which is completely divisible by 10, 16, and 24.
Answer:
the smallest square which divide 12,24 &40 is 3600
Step-by-step explanation:
first take LCM then
multiple you get 240 then
multiple 2×2×2×2×3×3×5×5 =3600
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