The number ${(101)}^{100} - 1$ is divisible by

10^{4}

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10^{6}

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10^{8}

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10^{12}

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Solution

The correct option is A $10^{4}$
Now ${(101)}^{100} - 1 = {(1 + 100)}^{100} - 1$

$= {^{100} C}_{1} 100 + {^{100} C}_{2} 100^{2} + . . . . . + {^{100} C}_{100} 100^{100}$

$= {(100)}^{2} [1 + {^{100} C}_{2} + {^{100} C}_{3} 1000 + . . .]$

Hence, it is divisible by $10^{4}$

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Similar questions

Q. The number

101^{100} - 1

is divisible by

Q. The value of

∣ ∣ ∣ ∣ \begin{matrix} 103 & 115 & 114 111 & 108 & 106 104 & 113 & 116 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} 113 & 116 & 104 108 & 106 & 111 115 & 114 & 103 \end{matrix} ∣ ∣ ∣ ∣

Q. Express the following numbers in usual form
(i)

3.02 \times 10^{- 6}

(ii)

4.5 \times 10^{4}

(iii)

3 \times 10^{- 8}

(iv)

1.0001 \times 10^{9}

(v)

5.8 \times 10^{12}

(vi)

3.61492 \times 10^{6}

Q. The value of sum of the given determinants,

| A | = ∣ ∣ ∣ ∣ \begin{matrix} 103 & 115 & 114 111 & 108 & 106 104 & 113 & 116 \end{matrix} ∣ ∣ ∣ ∣, | B | = ∣ ∣ ∣ ∣ \begin{matrix} 113 & 116 & 104 108 & 106 & 111 115 & 114 & 103 \end{matrix} ∣ ∣ ∣ ∣

Tick the correct answer:- In a homogeneous solution, the dimension of solute particles varies between:
a. 1 – 103 nm b. 104 - 106 nm
c. 105 - 106 nm d. 108 – 109 nm

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The number (101)100−1 is divisible by

The correct option is A 104Now (101)100−1=(1+100)100−1=100C1100+100C21002+.....+100C100100100=(100)2[1+100C2+100C31000+...]Hence, it is divisible by 104

The number ${(101)}^{100} - 1$ is divisible by

The correct option is A $10^{4}$
Now ${(101)}^{100} - 1 = {(1 + 100)}^{100} - 1$

$= {^{100} C}_{1} 100 + {^{100} C}_{2} 100^{2} + . . . . . + {^{100} C}_{100} 100^{100}$

$= {(100)}^{2} [1 + {^{100} C}_{2} + {^{100} C}_{3} 1000 + . . .]$

Hence, it is divisible by $10^{4}$