It is known that 11 divides both 33 and 121. Which statement is a direct consequence of the linear combination property of divisors?

Any multiple of 33 is divisible by 11

Divisors Practice — Grade 4 amc_math

Lesson 6: Divisors — Practice Questions

1. Since 7 divides both 28 and 63, the linear combination property states that 7 must also divide their sum. Their sum is ___.
2. The number 8 divides both 32 and 56. According to the property of linear combinations, which of the following numbers must also be divisible by 8?
- A. 32 + 56
- B. 56 / 32
- C. 32 * 56
- D. 56 - 30
3. If a number $k$ divides both integers $x$ and $y$, which of the following expressions is NOT guaranteed to be divisible by $k$ based on the property of linear combinations?
- A. x + y
- B. x - y
- C. y - x
- D. x + 1
4. We know that 4 divides both 16 and 28. The property of divisors states that 4 must also divide the difference $16 - 28$, which is ___.
5. It is known that 11 divides both 33 and 121. Which statement is a direct consequence of the linear combination property of divisors?
- A. 11 must divide 121 - 33
- B. 11 is a prime number
- C. 11 must divide 121 * 33
- D. Any multiple of 33 is divisible by 11
6. If 7 divides 21 and 21 divides 63, which conclusion is guaranteed by the transitivity property of divisors?
- A. $63 \mid 7$
- B. $7 \mid 63$
- C. $7 \mid 21$
- D. $3 \mid 7$
7. We know that 4 is a divisor of 20, and 20 is a divisor of 80. By the transitivity property, it must be true that 4 is a divisor of ___.
8. Which statement correctly demonstrates the transitivity property of divisors?
- A. Since $6 \mid 12$ and $12 \mid 36$, then $6 \mid 36$.
- B. Since $5 \mid 30$, then $30 = 5 \times 6$.
- C. Since $3 \mid 15$ and $3 \mid 21$, then $3 \mid (15+21)$.
- D. Since $10 = 2 \times 5$, then $2 \mid 10$ and $5 \mid 10$.
9. Let $m$, $n$, and $p$ be integers. If $m \mid n$ and $n \mid p$, then the transitivity property implies that $m \mid$ ___.
10. Given that $2 \mid 10$ and $10 \mid 50$, a student concludes that $2 \mid 50$. Is this conclusion a correct application of the transitivity property?
- A. Yes, the conclusion is valid.
- B. No, the conclusion is not valid.