Solveeit Logo
Login

Question

Question: If the given expression \[n\in N\],\[{{121}^{n}}-{{25}^{n}}+{{1900}^{n}}-{{\left( -4 \right)}^{n}}\]...

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

Explanation

Solution

Hint: To solve the question, we have to apply the formula that an−bn{{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.

Complete step-by-step answer:
We know that an−bn{{a}^{n}}-{{b}^{n}} is divisible by (a – b). By applying the formula we get

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

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

We know 96=16×6,1904=16×11996=16\times 6,1904=16\times 119

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

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

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

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

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

We know 1875=15×1251875=15\times 125

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

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

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

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

We know that product of 16 and 125 = 16×125=200016\times 125=2000

Thus, 2000 can divide the expression 121n−25n+1900n−(−4)n{{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 1900n−(−4)n{{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.