Simple applications of the binomial theorem — practice questions

Question 1 · difficulty L2 · understanding

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

• A. 4
• B. 6
• C. 3
• D. 1

Question 2 · difficulty L2 · understanding

Fractional part of the number $$\frac{4^{2022}}{15}$$ is equal to

• A. $$\frac{8}{15}$$
• B. $$\frac{4}{15}$$
• C. $$\frac{1}{15}$$
• D. $$\frac{14}{15}$$

Question 3 · difficulty L2 · understanding

Remainder when $$64^{32^{32}}$$ is divided by 9 is equal to ________.

Question 4 · difficulty L2 · understanding

The remainder, when $$7^{103}$$ is divided by 17, is __________

Question 5 · difficulty L2 · understanding

The largest natural number $$n$$ such that $$3^{n}$$ divides $$66 !$$ is ___________.

Question 6 · difficulty L2 · understanding

The remainder on dividing $$5^{99}$$ by 11 is ____________.

Question 7 · difficulty L3 · understanding

Given below are two statements : Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7. Statement II : The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number. In the light of the above statements, choose the correct answer from the options given below :

• A. Statement I is false but Statement II is true
• B. Both Statement I and Statement II are false
• C. Both Statement I and Statement II are true
• D. Statement I is true but Statement II is false

Question 8 · difficulty L3 · understanding

The largest $\mathrm{n} \in \mathbf{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is :

• A. 22
• B. 20
• C. 21
• D. 23

Question 9 · difficulty L3 · understanding

The remainder, when $7^{103}$ is divided by 23, is equal to:

• A. 9
• B. 6
• C. 14
• D. 17

Question 10 · difficulty L3 · understanding

Let the number $$(22)^{2022}+(2022)^{22}$$ leave the remainder $$\alpha$$ when divided by 3 and $$\beta$$ when divided by 7. Then $$\left(\alpha^{2}+\beta^{2}\right)$$ is equal to :

• A. 13
• B. 10
• C. 20
• D. 5