Understanding the Factorial Expression
The question asks for the smallest natural number \(n\) such that the expression \((n + 1) \times n \times (n - 1) \times (n - 2) \times ... \times 3 \times 2 \times 1\) is divisible by 910.
The expression \((n + 1) \times n \times (n - 1) \times ... \times 2 \times 1\) is the definition of the factorial of \((n+1)\), which is denoted as \((n+1)!\).
So, the core task is to find the smallest natural number \(n\) such that \((n+1)!\) is divisible by 910.
Prime Factorization of 910
To understand the divisibility requirement, we first find the prime factorization of the divisor, 910.
\(910 = 10 \times 91\)
Breaking down further:
- \(10 = 2 \times 5\)
- \(91 = 7 \times 13\)
Therefore, the prime factorization of 910 is:
\(910 = 2 \times 5 \times 7 \times 13\)
Divisibility Condition for Factorials
For a factorial, say \(k!\), to be divisible by another number, \(k!\) must contain all the prime factors of that number.
In our case, \((n+1)!\) must contain the prime factors 2, 5, 7, and 13.
The factorial \(k!\) includes all integers from 1 up to \(k\). For \((n+1)!\) to include the prime factors 2, 5, 7, and 13, the value of \((n+1)\) must be greater than or equal to the largest prime factor required.
The prime factors needed are 2, 5, 7, and 13. The largest among these is 13.
Thus, to ensure \((n+1)!\) has 13 as a factor, we must have:
\((n+1) \ge 13\)
Finding the Smallest Natural Number n
We are looking for the smallest natural number \(n\). This corresponds to the smallest possible value for \((n+1)\) that satisfies the condition \((n+1) \ge 13\).
The smallest integer value for \((n+1)\) that is greater than or equal to 13 is:
\((n+1) = 13\)
Now, we solve for \(n\):
\(n = 13 - 1\)
\(n = 12\)
Let's verify this. If \(n=12\), then \((n+1)! = 13!\). Since \(13! = 13 \times 12 \times ... \times 7 \times ... \times 5 \times ... \times 2 \times 1\), it definitely includes the factors 2, 5, 7, and 13. Therefore, \(13!\) is divisible by \(2 \times 5 \times 7 \times 13 = 910\).
If we chose \(n=11\), then \((n+1)! = 12!\). \(12!\) includes factors 2, 5, 7, but it does not include the factor 13. So, \(12!\) is not divisible by 910.
Therefore, the smallest natural number \(n\) is 12.
Similar Questions
- If \(n\) is a natural number, then what is the sum of all distinct remainders of \(4^n + 6^n + 9^n + 11^n\) when divided by 10 for various values of \(n\)?
- If \(a, b, c, d\) are natural numbers, then how many possible remainders are there when \(1^a + 2^b + 3^c + 4^d\) is divided by 10?
- The expression \(555^{777} + 777^{555}\) is divisible by which of the following?
1. 2
2. 3
3. 37
Select the correct answer using the code given below: - When the number (12345678910111213 ... 99100) is divided by 16, what will be the remainder?
- Consider the following statements :
1. If \((3m^3 + 2m^2 + 5m + n)/m\) is not an integer, where \(m\) and \(n\) are integers, then \(n\) is not divisible by \(m\).
2. \(5(8^m) + 2^{3m}\) is divisible by 48 for all whole numbers \(m\).
Which of the statements given above is/are correct? What is the remainder when \(111^{222} + 222^{333} + 333^{444}\) is divided by 5?
What is the remainder when \(3^{255}\) is divided by 28?
What is the value of \(x (0 \le x \le 8)\) if \((100^{97} + 100^{54} + x + 1)\) leaves a remainder 0 when divided by 9?
What is the remainder when
\(70 \times 71 \times 72 \times 73 \times 74 \times 75 \times 76 \times 77 \times 78 \times 79\) is divided by 1000 ?- What is the remainder if we divide \(3^{10}\) by 7?
Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
