Factors, multiples, prime and composite numbers Transitional Reasoning Practice

Questions

1. 1.

   A community center prepares care packages using 84 hygiene kits and 126 canned meals. Organizers want to pack all items into identical boxes so that each box receives the same number of hygiene kits and the same number of canned meals, leaving nothing unpacked. If they assemble the greatest possible number of identical boxes, what is the total number of items contained in each individual box?

   • A) 42
   • B) 14
   • C) 7
   • D) 5
2. 2.

   A digital lock requires a two-digit code $N$ to open a secure vault. The code $N$ is a composite number strictly between 40 and 80 whose greatest prime factor is 13. Furthermore, $N$ has exactly four distinct positive factors, and the sum of its digits is a prime number. What is the value of the code $N$?

   • A) 52
   • B) 78
   • C) 91
   • D) 65
3. 3.

   A baker bakes a batch of mini-tarts where the total count is greater than 70 and fewer than 100. When the tarts are packed into boxes of 4, 6, or 9, exactly 1 tart remains unpacked each time. Which of the following statements correctly describes the total number of mini-tarts baked?

   • A) It is a composite number that has exactly four distinct factors.
   • B) It is a prime number whose digits add up to 10.
   • C) It is a composite number that is a multiple of 7.
   • D) It is a prime number whose digits add up to 11.
4. 4.

   An auditorium coordinator is arranging chairs into equal rows for a seminar. The total number of chairs, $N$, is a two-digit composite number greater than 50 that is a multiple of 9. If $N$ has exactly 6 distinct positive factors and the sum of its digits is an even number, what is the value of $N$?

   • A) 63
   • B) 99
   • C) 81
   • D) 54
5. 5.

   A robotics club designs an access card that requires a two-digit secret key, $K$. The key $K$ is a multiple of 6 strictly less than 40, but it is not divisible by 4. Additionally, $K$ is a composite number that has exactly 8 distinct positive factors. Which number represents the secret key $K$?

   • A) 30
   • B) 24
   • C) 18
   • D) 36

Answers and step-by-step solutions appear after you finish the test.