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Stage 3 · Number theory

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Which of the following could not be the units digit [ones digit] of the square of a whole number?

   1. A1
   2. B4
   3. C5
   4. D6
   5. E8

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2. For how many integers $a$ with $1 \leq a \leq 10$ is $a^{2014}+a^{2015}$ divisible by 5?

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3. What is the ones digit of$$222,222-22,222-2,222-222-22-2?$$

   1. A0
   2. B2
   3. C4
   4. D6
   5. E8

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4. The largest desert in the world, the Sahara Desert in Africa, has a length of approximately $4800000m$. The number $4800000$ in scientific notation is represented as:

   1. A$48\times 10^{5}$
   2. B$4.8\times 10^{6}$
   3. C$0.48\times 10^{7}$
   4. D$4.8\times 10^{7}$

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5. First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is

   1. A$\frac{1}{16}$
   2. B$\frac{1}{8}$
   3. C$\frac{3}{16}$
   4. D$\frac{1}{5}$
   5. E$\frac{1}{4}$

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6. The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?

   1. A162
   2. B172
   3. C173
   4. D174
   5. E222

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7. Among the following numbers, the largest one is:

   1. A$0$
   2. B$\pi$
   3. C$\frac{1}{2023}$
   4. D$-2\pi$

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8. The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is

   1. A100
   2. B125
   3. C123
   4. D30
   5. E$\text{none of these}$

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9. How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?

   1. A0
   2. B1
   3. C2
   4. D3
   5. E4

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10. If $m$ and $n$ are positive integers with $n > 1$ such that $m^{n} = 2^{25} \times 3^{40}$, what is $m + n$?

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Answer key — Stage 3 · Number theory

Worked solutions for every problem are on the site, one page per problem.

1. $E$ open
2. $4$ open
3. $B$ open
4. $B$ open
5. $\frac{3}{16}$ open
6. $162$ open
7. $B$ open
8. $123$ open
9. $\textbf{(A)}0$ open
10. $209957$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.