If x is a natural number and 4 < x < 50, then the largest n, such that n! would always divide: $x (x^{2} - 1) (x^{2} - 4) (x^{2} - 9) (x + 4)$ is___

Open in App

Solution

The given expression can be written as $x (x^{2} - 1) (x^{2} - 4) (x^{2} - 9) (x + 4) = (x - 3) (x - 2) (x - 1) x (x + 1) (x + 2) (x + 3) (x + 4)$
It is the product of eight consecutive natural numbers so this product should be divisible by 8!. Hence, the largest n, would be n = 8

Suggest Corrections

Similar questions

Q. If A = {

x : x

is an even natural number} and B = {

y : y

is a natural number}, then

If U = {x: x is a natural number and x ≤ 10},
A = {a: a is an even natural number and a < 10}, and B = {b: b is a prime number and b ≤ 9}, then find A' ∩ B'.

Q. If a natural number

X

in the from of

Y^{2}

. If

Y

is also a natural number then

X

□

Q. {3, 4, 5, 6, 7} and {x: x is a natural number and 7≤ x ≤ 10} are sets.

Q. If

lim x \to 0 (1 + a x)^{\frac{b}{x}} = e^{4}

, where

a

and

b

are natural numbers, then

Join BYJU'S Learning Program

If x is a natural number and 4 < x < 50, then the largest n, such that n! would always divide: x(x2−1)(x2−4)(x2−9)(x+4) is___

The given expression can be written as x(x2−1)(x2−4)(x2−9)(x+4)=(x−3)(x−2)(x−1)x(x+1)(x+2)(x+3)(x+4) It is the product of eight consecutive natural numbers so this product should be divisible by 8!. Hence, the largest n, would be n = 8

If x is a natural number and 4 < x < 50, then the largest n, such that n! would always divide: $x (x^{2} - 1) (x^{2} - 4) (x^{2} - 9) (x + 4)$ is___

The given expression can be written as $x (x^{2} - 1) (x^{2} - 4) (x^{2} - 9) (x + 4) = (x - 3) (x - 2) (x - 1) x (x + 1) (x + 2) (x + 3) (x + 4)$
It is the product of eight consecutive natural numbers so this product should be divisible by 8!. Hence, the largest n, would be n = 8