Lesson 7: Greatest Common Divisor — Practice Questions

1. 1. To test if a number is divisible by 18, we can test for divisibility by two smaller numbers. Which pair of factors is valid for this test?

   • A. 2 and 9
   • B. 3 and 6
   • C. 1 and 18
   • D. 2 and 3

2. 2. The number 630 is divisible by 42. Which statement correctly explains why, using the divisibility property?

   • A. Because 630 is divisible by 2 and 21, and $\gcd(2, 21) = 1$.
   • B. Because 630 is divisible by 6 and 7, and $\gcd(6, 7) = 1$.
   • C. Because 630 is divisible by 3 and 14, and $\gcd(3, 14) = 1$.
   • D. All of the above.

3. 3. Knowing a number is divisible by 6 and 10 is not enough to conclude it is divisible by 60. This is because the factors 6 and 10 are not relatively prime; their greatest common divisor is ___.

4. 4. Which of the following statements is false?

   • A. If a number is divisible by 5 and 9, it must be divisible by 45.
   • B. If a number is divisible by 4 and 9, it must be divisible by 36.
   • C. If a number is divisible by 4 and 10, it must be divisible by 40.
   • D. If a number is divisible by 7 and 8, it must be divisible by 56.

5. 5. If the greatest common divisor of 8 and 12 is 4, what is the value of $\gcd(56, 84)$? The value is ___.

6. 6. For two positive integers $a$ and $b$, it is known that $\gcd(5a, 5b) = 30$. What is the value of $\gcd(a, b)$?

   • A. 5
   • B. 6
   • C. 30
   • D. 150

7. 7. Given that $\gcd(6, 10) = 2$ and $\gcd(24, 40) = 8$. What scaling factor $n$ makes the equation $\gcd(24, 40) = n \times \gcd(6, 10)$ true? The factor is ___.

8. 8. If two positive integers are both multiplied by a positive integer $k$, how does their greatest common divisor (GCD) change?

   • A. It is multiplied by k.
   • B. It is divided by k.
   • C. It is multiplied by k squared.
   • D. It does not change.

9. 9. Use the GCD Scaling Property to simplify and find the greatest common divisor of 150 and 210. The GCD is ___.