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What is the maximum possible power of 4 in the number that is obtained when the product of the first 15 positive integers is subtracted from the product of the first 20 positive integers?
- a)0
- b)3
- c)5
- d)7
- e)8
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
What is the maximum possible power of 4 in the number that is obtained...
Given
- Product of first 15 positive integers = 1*2*3…….* 15 = 15!
- Product of first 20 positive integers = 1*2*3*…….* 20 = 20!
- So, we can write 20! = 15! * 16 * 17 * 18 * 19 * 20
To Find: Maximum power of 4 that divides (20! – 15!)
Approach
- We will first need to simplify the expression 20! – 15!
- We know that 20! = 15! * 16 * 17 * 18 * 19 * 20
- So, 20! – 15! = 15! (20*19*18….*16 – 1) = 15! * odd integer
- Since 20 *19*…..16 is even, and 1 is odd, 20*19*……16 -1 will be odd
- Therefore, we only need to find the maximum power of 4 that divides 15!
2. To find the maximum power of 4 in 15!, we will first need to find the maximum power of 2 in 15!, as 4 = 22
- Let’s take a simple example to understand how we can find this. Consider a number p = 1*2*3*4. We need to find the power of 2 in p.
- Now, powers of 2 will occur in multiples of 2. So, we should first find the multiples of 2 in p. They are 2 and 4. However all multiples of 2 will not contain only 21. Some of them may contain higher powers of 2. For example, here 4 contains 22.
- So, when we are finding multiples of 2, we should also find the multiples of higher powers of 2 in p.
- Hence, 2 will occur 2 + 1 = 3 times in p.
3. We will use the same logic to find out the power of 2 in 15!
- Once we know the power of 2 in 15!, we can find the power of 4 in 15!
Working Out
Most Upvoted Answer
What is the maximum possible power of 4 in the number that is obtained...
To solve this problem, we need to find the maximum possible power of 4 in the difference between the product of the first 20 positive integers and the product of the first 15 positive integers.
Let's start by finding the product of the first 20 positive integers:
1 * 2 * 3 * ... * 18 * 19 * 20 = 2432902008176640000
Now let's find the product of the first 15 positive integers:
1 * 2 * 3 * ... * 13 * 14 * 15 = 1307674368000
To find the difference, we subtract the product of the first 15 positive integers from the product of the first 20 positive integers:
2432902008176640000 - 1307674368000 = 2432902006868972000
Now we need to find the maximum possible power of 4 in this difference. To do that, we need to find how many times we can divide the difference by 4 without getting a fraction.
Let's divide the difference by 4:
2432902006868972000 ÷ 4 = 608225501717243000
We can divide the result by 4 again:
608225501717243000 ÷ 4 = 152056375429310750
And once more:
152056375429310750 ÷ 4 = 38014093857327687.5
At this point, we can see that we have reached a fraction, so we can no longer divide by 4. Therefore, the maximum possible power of 4 in the difference is 3.
Therefore, the correct answer is option 'C', 5.
Let's start by finding the product of the first 20 positive integers:
1 * 2 * 3 * ... * 18 * 19 * 20 = 2432902008176640000
Now let's find the product of the first 15 positive integers:
1 * 2 * 3 * ... * 13 * 14 * 15 = 1307674368000
To find the difference, we subtract the product of the first 15 positive integers from the product of the first 20 positive integers:
2432902008176640000 - 1307674368000 = 2432902006868972000
Now we need to find the maximum possible power of 4 in this difference. To do that, we need to find how many times we can divide the difference by 4 without getting a fraction.
Let's divide the difference by 4:
2432902006868972000 ÷ 4 = 608225501717243000
We can divide the result by 4 again:
608225501717243000 ÷ 4 = 152056375429310750
And once more:
152056375429310750 ÷ 4 = 38014093857327687.5
At this point, we can see that we have reached a fraction, so we can no longer divide by 4. Therefore, the maximum possible power of 4 in the difference is 3.
Therefore, the correct answer is option 'C', 5.
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What is the maximum possible power of 4 in the number that is obtained when the product of the first 15 positive integers is subtracted from the product of the first 20 positive integers?a)0b)3c)5d)7e)8Correct answer is option 'C'. Can you explain this answer?
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