Prime and Composite

lessonbunny.com

Answer Key & Explanations

Prime and Composite · Grade 4 · Worksheet 3

1. Noah is organizing his baseball card collection into albums. He has 127 cards and wants to put the same number of cards on each page. If he uses 7 pages, will there be any cards left over? Explain how you know. Answer: Yes, there will be 1 card left over Solution: To determine if there will be any cards left over, we need to divide 127 by 7.
   Full step-by-step solution

   Step 1: To determine if there will be any cards left over, we need to divide 127 by 7. Step 2: 127 ÷ 7 = 18 with a remainder Step 3: 7 × 18 = 126 Step 4: 127 - 126 = 1 Step 5: Since we have a remainder of 1, this means there will be 1 card left over after putting 18 cards on each of the 7 pages. The answer is: Yes, there will be 1 card left over.

2. Noah is organizing his baseball card collection into albums. He has 127 cards and wants to put the same number of cards on each page. If he uses 7 pages, will he be able to place all cards equally with none left over? Explain why or why not. Answer: A. no Solution: We need to determine if 127 can be divided equally by 7 with no remainder. Calculate 127 ÷ 7 using long division. 7 × 18 = 126 127 - 126 = 1 Since there is a remainder of 1, 127 cannot be divided equally by 7.
   Full step-by-step solution

   Step 1: We need to determine if 127 can be divided equally by 7 with no remainder. Step 2: Calculate 127 ÷ 7 using long division. Step 3: 7 × 18 = 126 Step 4: 127 - 126 = 1 Step 5: Since there is a remainder of 1, 127 cannot be divided equally by 7. Step 6: Therefore, Noah cannot place all 127 cards equally on 7 pages with none left over.

3. A baker is arranging cupcakes on trays for a large order. She has 127 cupcakes and wants to arrange them in rows with the same number of cupcakes in each row. If she can only use arrangements where the number of cupcakes per row is greater than 1, how many different ways can she arrange the cupcakes? Answer: 1 Solution: We have 127 cupcakes. We want to arrange them in rows with the same number of cupcakes in each row. Number of cupcakes per row must be greater than 1.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have 127 cupcakes. We want to arrange them in rows with the same number of cupcakes in each row. Number of cupcakes per row must be greater than 1. We are asked: how many different ways can she arrange them? This means: find the number of positive divisors of 127 that are greater than 1, but also note that the arrangement means: If there are `r` cupcakes per row, then the number of rows is `127 / r`, which must be an integer. So `r` must be a divisor of 127. --- **Step 2: Check if 127 is prime** Test small primes: - Divisible by 2? No, it's odd. - Divisible by 3? Sum of digits = 1+2+7 = 10, not divisible by 3. - Divisible by 5? Last digit is 7, so no. - Divisible by 7? 7*18 = 126, remainder 1, so no. - Divisible by 11? 11*11 = 121, remainder 6, so no. - Divisible by 13? 13*9 = 117, remainder 10, no. - Divisible by 17? 17*7 = 119, remainder 8, no. - Next, 19*6=114, remainder 13, no. - 23*5=115, remainder 12, no. - We only need to check up to sqrt(127) ≈ 11.3, so we already checked all primes up to 11. Actually, sqrt(127) ≈ 11.27, so check primes ≤ 11: 2,3,5,7,11 — none divide 127. So 127 is prime. --- **Step 3: Divisors of 127** Since 127 is prime, its only positive divisors are 1 and 127. --- **Step 4: Apply the condition** We need number of cupcakes per row > 1. Possible divisors > 1: only 127. That means: - 127 cupcakes per row, 1 row. That is the only arrangement. --- **Step 5: Conclusion** Number of different arrangements = 1. --- **Final answer:** 1

4. Liam is organizing his rock collection into display cases. He has 127 rocks and wants to put the same number of rocks in each case. If he uses 7 cases, will there be any rocks left over? Explain how you know. Answer: Yes, there will be 1 rock left over Solution: Liam has 127 rocks and wants to divide them equally among 7 cases. Divide 127 by 7 to see if it divides evenly. 127 ÷ 7 = 18 with a remainder of 1 (because 7 × 18 = 126, and 127 - 126 = 1).
   Full step-by-step solution

   Step 1: Liam has 127 rocks and wants to divide them equally among 7 cases. Step 2: Divide 127 by 7 to see if it divides evenly. Step 3: 127 ÷ 7 = 18 with a remainder of 1 (because 7 × 18 = 126, and 127 - 126 = 1). Step 4: Since there is a remainder of 1, there will be 1 rock left over after putting 18 rocks in each of the 7 cases. The answer is: Yes, there will be 1 rock left over.

5. A rectangular garden is drawn with a length of 12 meters and a width of 8 meters. The gardener plants flowers in a square section that is 4 meters on each side, located in one corner of the garden. What is the area, in square meters, of the remaining part of the garden that is not planted with flowers? Answer: 80 Solution: Find the total area of the rectangular garden. The garden is a rectangle with length 12 m and width 8 m. Area of rectangle = length × width Total area = 12 × 8 = 96 square meters.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular garden. The garden is a rectangle with length 12 m and width 8 m. Area of rectangle = length × width Total area = 12 × 8 = 96 square meters. Step 2: Find the area of the square section planted with flowers. The square has side length 4 m. Area of square = side × side Flower area = 4 × 4 = 16 square meters. Step 3: Subtract the flower area from the total garden area to find the remaining area. Remaining area = Total area − Flower area Remaining area = 96 − 16 = 80 square meters. Step 4: Conclusion. The area of the remaining part of the garden is 80 square meters.

6. Aroha is sorting her collection of 73 seashells into display trays. She wants each tray to have the same number of shells, with no shells left over. She can use 1 tray for all 73 shells, or 73 trays with 1 shell each. Is 73 a prime or composite number? Explain why there are no other ways to arrange her shells into equal groups. Answer: prime Solution: A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Check if 73 is divisible by any number other than 1 and 73.
   Full step-by-step solution

   Step 1: A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Step 2: Check if 73 is divisible by any number other than 1 and 73. Step 3: Check 2: 73 is odd, so it is not divisible by 2. Step 4: Check 3: The sum of the digits of 73 is 7 + 3 = 10. 10 is not divisible by 3, so 73 is not divisible by 3. Step 5: Check 5: 73 does not end in 0 or 5, so it is not divisible by 5. Step 6: Check 7: 73 ÷ 7 = 10 with remainder 3, so it is not divisible by 7. Step 7: Check 11: 73 ÷ 11 = 6 with remainder 7, so it is not divisible by 11. Step 8: Since no number between 2 and 72 divides 73 evenly, 73 has only two factors: 1 and 73. Step 9: Therefore, 73 is a prime number. The answer is prime.