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Stage 4 · Number theory

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

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

1. How many two-digit squares differ by 1 from a multiple of 10 ?
   A 1
   B 2
   C 3
   D 4
   E 5

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2. A natural number $a$ has 107 different divisors, including 1 and $a$. Find the sum and product of these divisors.

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3. A3. How many divisors does the number 152 have?

   1. A7
   2. B8
   3. C9
   4. D10
   5. ENone of the above answers is correct

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4. Name a fraction greater than $\frac{11}{23}$ but less than $\frac{12}{23}$.

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5. What is the remainder when $(0!+1!+2!+\cdots+2011!)^{2}$ is divided by 10 ?

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6. Among the numbers $2019, 2021, 2023$, the number of prime numbers is $\qquad$ .

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7. If $2n+1, 20n+1 \left(n \in \mathbf{N}_{+}\right)$ are powers of the same positive integer, then all possible values of $n$ are

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8. Given the numbers $5^{1971}$ and $2^{1971}$. They are written consecutively. What is the number of digits in the resulting number?

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9. Find all integers $m, n, k$ greater than 1 such that
   $$1! + 2! + \cdots + m! = n^k.$$

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10. A four-digit number has all non-zero even digits, and its arithmetic square root is exactly a two-digit number, with both digits of this two-digit number also being non-zero even numbers. Then this four-digit number is $\qquad$

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Answer key — Stage 4 · Number theory

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

1. $2$ open
2. $\Sigma_{}=\frac{\sqrt[106]{}-1}{\sqrt[106]{}-1};\Pi_{}=\sqrt{}$ open
3. $8$ open
4. $\frac{1}{2}$ open
5. $6$ open
6. $1$ open
7. $4$ open
8. $1972$ open
9. $m=n=3$ open
10. $4624$ open

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