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16 Sep 2014, 01:22
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How many integers are divisible by 3 between \(10!\) and \(10! + 20\) inclusive?
A. 6
B. 7
C. 8
D. 9
E. 10
A. 6
B. 7
C. 8
D. 9
E. 10
16 Sep 2014, 01:22
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Official Solution:
How many integers are divisible by 3 between \(10!\) and \(10! + 20\) inclusive?
A. 6
B. 7
C. 8
D. 9
E. 10
Since 10! is a multiple of 3 (10! = 2 * 3 * ... * 10), the question essentially asks how many integers divisible by 3 are there between a multiple of 3 and that multiple of 3 + 20, inclusive.
Thus, we can rephrase the question as: how many integers are divisible by 3 from 0 to 20, inclusive?
We can easily count 7 such integers: 0, 3, 6, 9, 12, 15, and 18.
Alternatively, use the formula to find the number of multiples of \(x\) in a range: \(\frac{\text{Last multiple of x in the range - First multiple of x in the range}}{x}+1\).
So, \(\frac{18-0}{3}+1=7\).
Therefore, there are 7 integers divisible by 3 between \(10!\) and \(10! + 20\), inclusive.
Answer: B
How many integers are divisible by 3 between \(10!\) and \(10! + 20\) inclusive?
A. 6
B. 7
C. 8
D. 9
E. 10
Since 10! is a multiple of 3 (10! = 2 * 3 * ... * 10), the question essentially asks how many integers divisible by 3 are there between a multiple of 3 and that multiple of 3 + 20, inclusive.
Thus, we can rephrase the question as: how many integers are divisible by 3 from 0 to 20, inclusive?
We can easily count 7 such integers: 0, 3, 6, 9, 12, 15, and 18.
Alternatively, use the formula to find the number of multiples of \(x\) in a range: \(\frac{\text{Last multiple of x in the range - First multiple of x in the range}}{x}+1\).
So, \(\frac{18-0}{3}+1=7\).
Therefore, there are 7 integers divisible by 3 between \(10!\) and \(10! + 20\), inclusive.
Answer: B
11 Jul 2015, 02:49
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vikasbansal227
The solution does NOT say that 10! contains one multiple of 3, it says that 10! IS a multiple of 3, it does not matter what is the power of 3 in 10!.
Multiples of 3 between \(10!\) and \(10! + 20\) inclusive are:
\(10!\);
\(10! + 3\);
\(10! + 6\);
\(10! + 9\);
\(10! + 12\);
\(10! + 15\);
\(10! + 18\).
Total of 7.
