Numbers & Operations — Quiz 4

Questions in this quiz (11)

1. If $$a$$ and $$b$$ are positive integers such that $$a$$ < $$b$$, which of the following could be irrational?

   • $$\sqrt{ab}$$
   • $$a + b$$
   • $$b - a$$
   • $$\frac{a}{b}$$

2. For all nonzero values of $$a$$ and $$b$$, which of the following expressions will never equal zero?

   • $$|a+b|$$
   • $$|a|+|b|$$
   • $$|a|-|b|$$
   • $$a-b$$

3. Given that $$y$$ and $$\frac{7-y}{2}$$ are integers, which of the following statements about $$y$$ must be true?

   • $$y$$ is odd
   • $$y$$ is even
   • $$y$$ is divisible by 7
   • $$y$$ is negative

4. Suppose that $$y$$ is a real number and $$\frac{1}{y}$$ is a rational number. Which of the following could be the value of $$y$$?I. $$0$$II. $$\sqrt{2}$$III. $$-\frac{1}{2}$$

   • III only
   • I and II only
   • II and III only
   • I, II, and III

5. The symbol $$\#$$ is defined such that:$$a\#b=a^2+b^2-2ab$$If $$x\#3=\frac{4x-6}{7}$$, find the value of $$x$$ given that $$x\neq3.$$

   • $$\frac{9\pm\sqrt{3}}{4}$$
   • $$\frac{-81\pm\sqrt{3}}{4}$$
   • $$\frac{81\pm\sqrt{3}}{4}$$
   • $$\frac{46\pm\sqrt{184}}{14}$$

6. If $$x, y, z$$ are single-digit integers and$$M=24000\times x+100\times y+10\times z$$.What is the digit in the hundreds place of number $$M$$?

   • $$0$$
   • $$x$$
   • $$y$$
   • $$z$$

7. A three-digit number, $$ABC$$, is formed using three different nonzero digits $$A,B,C$$. A new number is created by rearranging the same three digits. What is the greatest possible difference between the two numbers?

   • $$198$$
   • $$253$$
   • $$157$$
   • $$456$$

8. A particular integer $$M$$ is divisible by three different prime numbers $$p,q,$$ and $$r$$. Which of the following must be true?I. $$M$$ is not a prime number.II. $$M$$ is divisible by $$pqr$$.III. $$M$$ is an even integer.

   • I, III
   • II, III
   • I, II
   • I, II, III

9. The square of a positive integer cannot be of the form (where $$k$$ is an integer):

   • $$3k + 1$$
   • $$5k + 4$$
   • $$4k + 2$$
   • $$5k + 1$$

10. If $$a = 2^p \cdot 3^q \cdot 5^r$$ and $$b = p^2 \cdot q^3 \cdot r^5$$, where $$p$$, $$q$$ and $$r$$ are primes, and $$a = b$$, find the value of $$p + q + r$$.

    • 5
    • 10
    • 7
    • 11

11. If the sum of a positive number $$x$$ and its reciprocal is 2, then find the value of $$x^{10}+\frac{1}{x^{10}}$$.

    • $$2^{10}$$
    • $$\frac{1}{2^{10}}$$
    • $$20$$
    • $$2$$