Prime and Composite

lessonbunny.com

Answer Key & Explanations

Prime and Composite · Grade 4 · Worksheet 1

1. Noah is helping his teacher organize the classroom library. He has 53 books to put on shelves. He wants to arrange them so that each shelf has the same number of books, with no books left over. Can Noah arrange the 53 books into equal groups on shelves? Explain why 53 is a prime or composite number. Answer: 53 is a prime number Solution: A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Check if 53 is divisible by 2.
   Full step-by-step solution

   Step 1: Understand the definitions. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Step 2: Check if 53 is divisible by 2. 53 is odd, so it is not divisible by 2. Step 3: Check if 53 is divisible by 3. Add the digits: 5 + 3 = 8. 8 is not divisible by 3, so 53 is not divisible by 3. Step 4: Check if 53 is divisible by 5. 53 does not end in 0 or 5, so it is not divisible by 5. Step 5: Check if 53 is divisible by 7. 7 x 7 = 49 and 7 x 8 = 56. 53 is not a multiple of 7. Step 6: The only possible divisors to check are 2, 3, 5, and 7 because any larger factor would need a smaller partner factor, and we've already checked all smaller numbers. None of these divide 53 evenly. Step 7: Therefore, 53 has only two factors: 1 and 53. This means 53 is a prime number. Step 8: Since 53 is prime, Noah cannot arrange the books into equal groups on shelves (other than 1 shelf with 53 books or 53 shelves with 1 book each). The answer is 53 is a prime number.

2. Noah is drawing a number chart on a 6 by 6 grid. He shades all the prime numbers between 1 and 36. How many numbers does Noah shade? Answer: 11 Solution: List all numbers from 2 to 36. Step 2: Identify the prime numbers in that range. A prime number has only two factors: 1 and itself.
   Full step-by-step solution

   Step 1: List all numbers from 2 to 36. Step 2: Identify the prime numbers in that range. A prime number has only two factors: 1 and itself. Step 3: The prime numbers between 2 and 36 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. Step 4: Count them: there are 11 numbers. Step 5: So Noah shades 11 numbers. The answer is 11.

3. Matiu is looking at a grid of numbers. The grid has numbers arranged in rows and columns. In one row, he sees the numbers 24, 31, 42, and 53. Matiu wants to color all the composite numbers in that row blue and all the prime numbers red. How many numbers in the row will he color blue? Answer: 2 Solution: List the numbers: 24, 31, 42, 53. Check each number to see if it is prime or composite. - 24: Factors are 1, 2, 3, 4, 6, 8, 12, 24.
   Full step-by-step solution

   Step 1: List the numbers: 24, 31, 42, 53. Step 2: Check each number to see if it is prime or composite. - 24: Factors are 1, 2, 3, 4, 6, 8, 12, 24. It has more than two factors, so it is composite. - 31: Factors are 1 and 31. It has exactly two factors, so it is prime. - 42: Factors are 1, 2, 3, 6, 7, 14, 21, 42. It has more than two factors, so it is composite. - 53: Factors are 1 and 53. It has exactly two factors, so it is prime. Step 3: Count the composite numbers (the ones to color blue). The composite numbers are 24 and 42. Step 4: There are 2 composite numbers. The answer is 2.

4. Sophia is drawing a number line from 0 to 100 on a long strip of paper. She marks a red dot on the number 23 and a blue dot on the number 67. She then says, 'Both of these numbers are prime.' Is Sophia correct? Explain whether 23 and 67 are prime or composite numbers. Answer: Both 23 and 67 are prime numbers. Solution: Check 23. Try dividing 23 by 2: 23 ÷ 2 = 11.5 (not whole). Try 3: 23 ÷ 3 = 7.666...
   Full step-by-step solution

   Step 1: Check 23. Try dividing 23 by 2: 23 ÷ 2 = 11.5 (not whole). Try 3: 23 ÷ 3 = 7.666... (not whole). Try 5: 23 ÷ 5 = 4.6 (not whole). Try 7: 7 × 3 = 21, 7 × 4 = 28 (too big), so no. Try 11: 11 × 2 = 22, 11 × 3 = 33 (too big). Since no whole number other than 1 and 23 divides 23, it has exactly two factors: 1 and 23. So 23 is prime. Step 2: Check 67. Try dividing 67 by 2: 67 ÷ 2 = 33.5 (not whole). Try 3: 67 ÷ 3 = 22.333... (not whole). Try 5: 67 ÷ 5 = 13.4 (not whole). Try 7: 7 × 9 = 63, 7 × 10 = 70 (too big), so no. Try 11: 11 × 6 = 66, 11 × 7 = 77 (too big). Since no whole number other than 1 and 67 divides 67, it has exactly two factors: 1 and 67. So 67 is prime. Step 3: Conclusion. Both 23 and 67 are prime numbers, so Sophia is correct.

5. Emma is organizing her rock collection into display cases. She has 127 rocks and wants to arrange them into equal rows. If she tries to make 7 equal rows, will she have any rocks left over? Explain how you know. Answer: Yes, she will have 1 rock left over Solution: Emma has 127 rocks and wants to make 7 equal rows To check if 127 is divisible by 7, we can divide 127 by 7 7 × 18 = 126 127 - 126 = 1 Since there is a remainder of 1, Emma will have 1 rock left over after making 7 equal rows Each row would have 18 rocks, and 1 rock would remain Therefore, Emma…
   Full step-by-step solution

   Step 1: Emma has 127 rocks and wants to make 7 equal rows Step 2: To check if 127 is divisible by 7, we can divide 127 by 7 Step 3: 7 × 18 = 126 Step 4: 127 - 126 = 1 Step 5: Since there is a remainder of 1, Emma will have 1 rock left over after making 7 equal rows Step 6: Each row would have 18 rocks, and 1 rock would remain Therefore, Emma will have 1 rock left over.

6. Emma is organizing her sticker collection into groups. She has 47 stickers and wants to arrange them into equal rows. Can she make 2 equal rows? Can she make 3 equal rows? Can she make 5 equal rows? For which of these numbers can she arrange all her stickers with none left over? Answer: Only 47 Solution: Emma has 47 stickers. She wants to arrange them into equal rows, with no stickers left over. We check if she can make 2, 3, or 5 equal rows.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understanding the problem** Emma has 47 stickers. She wants to arrange them into equal rows, with no stickers left over. We check if she can make 2, 3, or 5 equal rows. That means: check if 47 is divisible by 2, 3, and 5. --- **Step 2: Check divisibility by 2** A number is divisible by 2 if it is even (last digit is 0, 2, 4, 6, or 8). 47 ends with 7 → odd → not divisible by 2. So: 47 ÷ 2 = 23 remainder 1. She cannot make 2 equal rows. --- **Step 3: Check divisibility by 3** A number is divisible by 3 if the sum of its digits is divisible by 3. Digits of 47: 4 + 7 = 11. 11 is not divisible by 3 (11 ÷ 3 = 3 remainder 2). So 47 is not divisible by 3. 47 ÷ 3 = 15 remainder 2. She cannot make 3 equal rows. --- **Step 4: Check divisibility by 5** A number is divisible by 5 if it ends in 0 or 5. 47 ends with 7 → not divisible by 5. 47 ÷ 5 = 9 remainder 2. She cannot make 5 equal rows. --- **Step 5: Conclusion** Only if the number of rows divides 47 evenly can she arrange all stickers with none left over. Since 47 is prime, its only divisors are 1 and 47. So among the choices 2, 3, 5 — none work. The only way to have equal rows with no leftovers is 1 row of 47 stickers or 47 rows of 1 sticker. Thus, the correct answer is: **Only 47** (meaning 47 equal rows of 1 sticker each, or 1 row of 47 stickers). --- **Final Answer:** Only 47

7. Aroha is helping her teacher arrange chairs for a class performance. They have 59 chairs to set up in equal rows. Can Aroha arrange all 59 chairs into equal rows of more than one row and less than 59 rows, with no chairs left over? Explain whether 59 is a prime or composite number. Answer: 59 is a prime number. Solution: We need to determine if 59 is prime or composite. A prime number has only two factors: 1 and itself. A composite number has more than two factors.
   Full step-by-step solution

   Step 1: We need to determine if 59 is prime or composite. A prime number has only two factors: 1 and itself. A composite number has more than two factors. Step 2: Check if 59 is divisible by any number other than 1 and 59. - Check 2: 59 is odd, so not divisible by 2. - Check 3: 5 + 9 = 14, and 14 is not divisible by 3, so 59 is not divisible by 3. - Check 5: 59 does not end in 0 or 5, so not divisible by 5. - Check 7: 7 x 8 = 56, 7 x 9 = 63. 59 is not a multiple of 7. - Check 11: 11 x 5 = 55, 11 x 6 = 66. 59 is not a multiple of 11. - Check 13: 13 x 4 = 52, 13 x 5 = 65. 59 is not a multiple of 13. - Check 17: 17 x 3 = 51, 17 x 4 = 68. 59 is not a multiple of 17. - Check 19: 19 x 3 = 57, 19 x 4 = 76. 59 is not a multiple of 19. - Check 23: 23 x 2 = 46, 23 x 3 = 69. 59 is not a multiple of 23. - Check 29: 29 x 2 = 58, 29 x 3 = 87. 59 is not a multiple of 29. - Since 7 x 7 = 49 and 11 x 11 = 121, we only need to check prime numbers up to 7 (since 7 x 7 = 49 is less than 59, but 11 x 11 = 121 is greater). We checked 2, 3, 5, 7 and none divide 59 evenly. Step 3: Since no number other than 1 and 59 divides evenly into 59, 59 has exactly two factors: 1 and 59. Step 4: Therefore, 59 is a prime number. Aroha cannot arrange the chairs into equal rows of more than one row and less than 59 rows with no chairs left over. The answer is 59 is a prime number.

8. A rectangular swimming pool is drawn on a grid. The pool measures 15 meters in length and 10 meters in width. A square diving area in one corner of the pool has sides measuring 4 meters. What is the area of the pool that is available for swimming, in square meters? Answer: 134 Solution: Find the total area of the rectangular pool. Area of a rectangle = length × width. So, 15 × 10 = 150 square meters.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular pool. Area of a rectangle = length × width. So, 15 × 10 = 150 square meters. Step 2: Find the area of the square diving area. Area of a square = side × side. So, 4 × 4 = 16 square meters. Step 3: Subtract the area of the diving area from the total pool area to find the swimming area. 150 - 16 = 134 square meters. The area available for swimming is 134 square meters.