Question

elating factors and multiples i - item 33494
amys grandmother is exactly 6 times older than amy.
which three statements below must be true?
the age of amys grandmother is a prime number. the age of amys grandmother is a composite number.
amys age is a factor of the age of her grandmother. the number 6 is a factor of the age of amys grandmother.
the age of amys grandmother has exactly two factors. the age of amys grandmother has exactly four factors.

Explanation:

Step1: Analyze Prime/Composite

Let Amy's age be \( n \), grandmother's age is \( 6n \). A prime number has exactly 2 factors (1 and itself). \( 6n \) has factors like 1, 2, 3, 6, \( n \), \( 2n \), \( 3n \), \( 6n \) (for \( n>1 \)), so it's composite (has more than 2 factors). So "grandmother's age is composite" is true, "prime" is false.

Step2: Factor Relationship

If grandmother's age is \( 6n \), then \( n \) (Amy's age) divides \( 6n \), so Amy's age is a factor. Also, 6 divides \( 6n \), so 6 is a factor.

Step3: Number of Factors

Composite numbers can have different factor counts. \( 6n \) (composite) doesn't necessarily have exactly 2 or 4 factors. E.g., if \( n = 2 \), \( 6n = 12 \), factors: 1,2,3,4,6,12 (6 factors). So "exactly two" and "exactly four" are false.

Answer:

The three true statements are:

• The age of Amy’s grandmother is a composite number.
• Amy’s age is a factor of the age of her grandmother.
• The number 6 is a factor of the age of Amy’s grandmother.