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Find the maximum value of n such that 570 X 60 X 30 X 90 X 100 X 500 X 700 X 343 X 720 X 81 is perfectly divisible by 30n.
- a)12
- b)11
- c)14
- d)13
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Find the maximum value of n such that 570 X 60 X 30 X 90 X 100 X 500 X...
Checking for the number of 2’s, 3’s and 5’s in the given expression you can see that the minimum is for the number of 3’s (there are 11 of them while there are 12 5’s and more than 11 2’s) Hence, option (b) is correct.
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Find the maximum value of n such that 570 X 60 X 30 X 90 X 100 X 500 X...
To find the maximum value of n, we need to determine the highest power of 30 that divides the given expression.
Given expression: 570 X 60 X 30 X 90 X 100 X 500 X 700 X 343 X 720 X 81
Breaking down the expression:
570 = 2 X 3 X 5 X 19
60 = 2^2 X 3 X 5
30 = 2 X 3 X 5
90 = 2 X 3^2 X 5
100 = 2^2 X 5^2
500 = 2^2 X 5^3
700 = 2^2 X 5^2 X 7
343 = 7^3
720 = 2^4 X 3^2 X 5
81 = 3^4
Now, let's calculate the power of 30 in the given expression:
Power of 2 = 4 + 1 + 1 + 1 + 2 + 2 + 4 = 15
Power of 3 = 1 + 1 + 2 + 4 = 8
Power of 5 = 1 + 1 + 1 + 2 + 3 = 8
Power of 7 = 3
To find the power of 30, we take the minimum power among the prime factors 2, 3, 5, and 7. In this case, the power of 2, 3, and 5 is 8, and the power of 7 is 3. Therefore, the maximum power of 30 is 8.
Since the given expression is perfectly divisible by 30n, the maximum value of n would be the power of 30, which is 8.
Hence, the correct answer is option B) 11.
Given expression: 570 X 60 X 30 X 90 X 100 X 500 X 700 X 343 X 720 X 81
Breaking down the expression:
570 = 2 X 3 X 5 X 19
60 = 2^2 X 3 X 5
30 = 2 X 3 X 5
90 = 2 X 3^2 X 5
100 = 2^2 X 5^2
500 = 2^2 X 5^3
700 = 2^2 X 5^2 X 7
343 = 7^3
720 = 2^4 X 3^2 X 5
81 = 3^4
Now, let's calculate the power of 30 in the given expression:
Power of 2 = 4 + 1 + 1 + 1 + 2 + 2 + 4 = 15
Power of 3 = 1 + 1 + 2 + 4 = 8
Power of 5 = 1 + 1 + 1 + 2 + 3 = 8
Power of 7 = 3
To find the power of 30, we take the minimum power among the prime factors 2, 3, 5, and 7. In this case, the power of 2, 3, and 5 is 8, and the power of 7 is 3. Therefore, the maximum power of 30 is 8.
Since the given expression is perfectly divisible by 30n, the maximum value of n would be the power of 30, which is 8.
Hence, the correct answer is option B) 11.
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Find the maximum value of n such that 570 X 60 X 30 X 90 X 100 X 500 X 700 X 343 X 720 X 81 is perfectly divisible by 30n.a)12b)11c)14d)13Correct answer is option 'B'. Can you explain this answer?
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