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# If the given expression $n\in N$,${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$ then is divisible by which one of the following?a)1904b)2000c)2002d)2006

Last updated date: 17th Mar 2023
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Hint: To solve the question, we have to apply the formula that ${{a}^{n}}-{{b}^{n}}$ is divisible by (a – b). Apply the formula to all terms of the expression to find common divisible factors of the expression.

We know that ${{a}^{n}}-{{b}^{n}}$ is divisible by (a – b). By applying the formula we get

${{121}^{n}}-{{25}^{n}}$ is divisible by (121 - 25) = 96

${{1900}^{n}}-{{\left( -4 \right)}^{n}}$is divisible by (1900 – (-4)) = 1900 + 4 = 1904

We know $96=16\times 6,1904=16\times 119$

Thus, the common factor of 96, 1904 is 16.

Thus, 16 can divide the expression ${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$

By applying the above formula for another set of terms of expression, we get

${{121}^{n}}-{{\left( -4 \right)}^{n}}$is divisible by (121 – (-4)) = 121 + 4 = 125

${{1900}^{n}}-{{25}^{n}}$is divisible by (1900 - 25) = 1875

We know $1875=15\times 125$

Thus, the common factor of 125, 1875 is 125.

Thus, 125 can divide the expression ${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$

Thus, we get both 16 and 125can divide the expression ${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$

This implies that the product of 16 and 125 can divide the expression ${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$

We know that product of 16 and 125 = $16\times 125=2000$

Thus, 2000 can divide the expression ${{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}$

Hence, option (b) is the right answer.

Note: The possibility of mistake can be interpreted that 1904 divides the given expression because it divides ${{1900}^{n}}-{{\left( -4 \right)}^{n}}$. But it is not divisible by the other part of the expression, only common factors can divide the whole expression. The alternative to solve the questions is equal to substitute n = 1 in the given expression, the calculated value is equal to 2000. Hence, the other options can be eliminated.