Question

relating factors and multiples i - item 33578
question 9 of 10
carls grandfather is exactly 5 times older than carl.
which three statements must be true?
the age of carls grandfather is a prime number.
the age of carls grandfather is a composite number.
the age of carls grandfather has exactly two factors.
the age of carls grandfather has exactly four factors.
carls age is a factor of the age of his grandfather.
the number 5 is a factor of the age of carls grandfather.

Explanation:

• Let Carl’s age = \( C \), grandfather’s age = \( 5C \). \( 5C \) has factors 1, 5, \( C \), \( 5C \) (at least 4 factors if \( C

eq 1 \), which is realistic for age), so it’s composite (not prime, which has exactly 2 factors).

• Since \( 5C \div C = 5 \), Carl’s age is a factor.
• \( 5C \div 5 = C \), so 5 is a factor.
• "Exactly four factors" isn’t guaranteed (e.g., if \( C=5 \), grandfather=25 has factors 1,5,25: 3 factors).
• Prime/2 factors are false as \( 5C \) is composite.

Thus the three true statements are: The age of Carl’s grandfather is a composite number; Carl’s age is a factor of the age of his grandfather; The number 5 is a factor of the age of Carl’s grandfather.

Answer:

The age of Carl’s grandfather is a composite number, Carl’s age is a factor of the age of his grandfather, The number 5 is a factor of the age of Carl’s grandfather