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The greatest natural number, which exactly divides the product of any four consecutive natural numbers is:
120
6
24
12
24
The product of k consecutive natural numbers is always divisible by k!. For 4 numbers, the divisor is 4!=4├Ч3├Ч2├Ч1=24.
The product of any four consecutive natural numbers.
P=n(n+1)(n+2)(n+3)
The product of k consecutive natural numbers is always divisible by k!. For 4 numbers, the divisor is 4!=4├Ч3├Ч2├Ч1=24.
Students often mistake this for 5!=120, but 120 is the result for 5 consecutive natural numbers.
Define the expression
Let the four consecutive natural numbers be n,n+1,n+2, and n+3. Their product is P=n(n+1)(n+2)(n+3).
P=n(n+1)(n+2)(n+3)
Apply the rule of consecutive integers
The product of k consecutive natural numbers is always divisible by the factorial of k (k!). Here k=4.
4!=4├Ч3├Ч2├Ч1=24
Verify with an example
Take the first four natural numbers: 1,2,3,4. Their product is 1├Ч2├Ч3├Ч4=24. Since 24 divides 24, this is the greatest common divisor for any sequence.
24├╖24=1
C is correct because the product of any 4 consecutive natural numbers is always exactly divisible by 4!=24.
This is a specific case of the broader property that the product of n consecutive integers is always divisible by n!.
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