Quiz: Number Systems Practice Problems for Exams (set 1)

This quiz is designed to test your understanding of number theory through a variety of questions that emphasize conceptual knowledge, creativity, and problem-solving skills. It covers a range of topics, including divisibility, prime numbers, number patterns, and algebraic equations. Anyone interested in practicing these concepts is welcome to participate. Try to solve the questions without external help. If you don’t get all the answers correct on your first attempt, don’t worry—you can retake the quiz to aim for a perfect score. If you notice any errors in the questions or answers, please leave a comment so we can address them. Enjoy the quiz and good luck!

1. 
The sum of three consecutive integers is 72. Find the largest of the three integers.

2. 
The sum of two consecutive odd numbers is 56. Find the smallest of the two numbers.

3. 
A number when divided by 5 leaves a remainder of 3, and when divided by 7 leaves a remainder of 2. Find the smallest such number.

4. 
If a number $N$ has the prime factorization $2^3 \times 3^2 \times 5$, find the total number of divisors of $N$.

5. 
If a number $N$ has the prime factorization $2^3 \times 3^2 \times 5$, find the sum of all divisors of $N$.

6. 
A two-digit number is such that the sum of its digits is 9. If the digits are reversed, the new number is 27 less than the original number. Find the original number.

7. 
The sum of the first $n$ odd numbers is $n^2$. Use this fact to find the sum of the first 20 odd numbers.

8. 
A number is 5 less than twice another number. If their sum is 25, find the two numbers.

9. 
A number squared is equal to 4 times the number plus 12. Find the number(s).

10. 
A rectangle has a length that is 3 more than twice its width. If the area of the rectangle is 65 square units, find the perimeter of the rectangle.

11. 
A two-digit number is such that the product of its digits is 12. If the digits are reversed, the new number is 9 less than the original number. Find the original number.

12. 
If $GCD(a,b)=4$ and $LCM(a,b)=48$, find the smallest possible values of $a$ and $b$.

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