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# AOPS Help

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1.) Compute $14A6_{12} - 5B9_{12}$. Give your answer as a base 12 integer.

(You do not need to include the subscript to indicate the base.)

2.) Compute $101_2 + 11_2 + 1100_2 + 11101_2$. Give your answer as a base 2 integer.

(You do not need to include the subscript to indicate the base.)

3.) Compute $4392_{11} - 14A3_{11} + 13A_{11}$. Give your answer as a base 11 integer.

(You do not need to include the subscript to indicate the base.)

4.) Compute $43_6 \cdot 51_6$ in base 6.

5.) Compute $12121_3 \cdot 1222_3$ in base 3.

6.) Compute $1980_{16}/18_{16}$ in base 16.

7.) Express $0.\overline{21}_3$ as a base 10 fraction in reduced form.

8.) What is $1000000_{16} - 8D6A7B_{16}$ in base 16?

(You do not need to include the subscript 16 in your answer.)

9.)

Find a base 7 three-digit number which has its digits reversed when expressed in base 9.

(You do not need to indicate the base with a subscript for this answer.)

Hint(s):

Numbers written in base 7 can only use the digits 0, 1, 2, 3, 4, 5, or 6.

10.) Express $3.\overline{4}_{13}$ as a base 10 fraction in reduced form.

A.) Jon teaches a fourth grade class at an elementary school where class sizes are always at least 20 students and at most 28. One day Jon decides that he wants to arrange the students in their desks in a rectangular grid with no gaps. Unfortunately for Jon he discovers that doing so could only result in one straight line of desks. How many students does Jon have in his class?

B.) How many bases b are there such that $663_b$ is prime?

C.) Find the greatest prime divisor of the sum of the arithmetic sequence $1 + 2 + 3 + \dots + 80.$

Jun 29, 2019