Prime and Composite

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Answer Key & Explanations

Prime and Composite · Grade 4 · Worksheet 2

1. A rectangular swimming pool is drawn on a grid. The pool is 15 meters long and 10 meters wide. A square diving area with sides of 4 meters is marked off in one corner of the pool. What is the area, in square meters, of the swimming pool that is NOT part of the diving area? Answer: 134 Solution: Area of rectangle = length × width = 15 × 10 = 150 square meters Area of square = side × side = 4 × 4 = 16 square meters 150 - 16 = 134 square meters The area of the swimming pool that is NOT part of the diving area is 134 square meters.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular pool Area of rectangle = length × width = 15 × 10 = 150 square meters Step 2: Find the area of the square diving area Area of square = side × side = 4 × 4 = 16 square meters Step 3: Subtract the diving area from the total pool area 150 - 16 = 134 square meters The area of the swimming pool that is NOT part of the diving area is 134 square meters.

2. Emma is drawing a number grid on a large piece of paper. The grid has 10 rows and 10 columns, with numbers from 1 to 100 arranged in order. She circles all the prime numbers in the grid. Look at the fifth row: 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. How many numbers in this row does Emma circle? Answer: 3 Solution: List the numbers in the fifth row: 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. Check each number to see if it is prime. - 41: 41 divided by 2 is 20.5, by 3 is 13.666, by 5 is 8.2, by 7 is 5.857.
   Full step-by-step solution

   Step 1: List the numbers in the fifth row: 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. Step 2: Check each number to see if it is prime. - 41: 41 divided by 2 is 20.5, by 3 is 13.666, by 5 is 8.2, by 7 is 5.857. No whole number divides it evenly except 1 and 41. So 41 is prime. - 42: Even number, divisible by 2. Composite. - 43: 43 divided by 2 is 21.5, by 3 is 14.333, by 5 is 8.6, by 7 is 6.142. No whole number divides it evenly except 1 and 43. So 43 is prime. - 44: Even number, divisible by 2. Composite. - 45: 45 divided by 5 is 9, so it has factors 1, 5, 9, 45. Composite. - 46: Even number, divisible by 2. Composite. - 47: 47 divided by 2 is 23.5, by 3 is 15.666, by 5 is 9.4, by 7 is 6.714. No whole number divides it evenly except 1 and 47. So 47 is prime. - 48: Even number, divisible by 2. Composite. - 49: 49 divided by 7 is 7, so it has factors 1, 7, 49. Composite. - 50: Even number, divisible by 2. Composite. Step 3: The prime numbers in this row are 41, 43, and 47. That is 3 numbers. The answer is 3.

3. Is 37 a prime number? Answer: yes Solution: A prime number has only two factors: 1 and itself. Step 2: Find the square root of 37. sqrt(37) is about 6.08.
   Full step-by-step solution

   Step 1: A prime number has only two factors: 1 and itself. Step 2: Find the square root of 37. sqrt(37) is about 6.08. So we only need to test prime numbers less than or equal to 6: 2, 3, and 5. Step 3: Test divisibility by 2. 37 is odd, so it is not divisible by 2. Step 4: Test divisibility by 3. Sum of digits: 3 + 7 = 10. 10 is not divisible by 3, so 37 is not divisible by 3. Step 5: Test divisibility by 5. The last digit of 37 is 7, not 0 or 5, so it is not divisible by 5. Step 6: Since no prime number less than or equal to sqrt(37) divides 37 evenly, 37 has no factors other than 1 and itself. Therefore, 37 is a prime number. The answer is yes.

4. Kaia is helping her teacher arrange the classroom chairs for a special assembly. She has 53 chairs to set up in equal rows. Can Kaia arrange all 53 chairs into 5 equal rows with no chairs left over, or into 7 equal rows with no chairs left over? Explain whether 53 is a prime or composite number and why. Answer: 53 is a prime number. It cannot be arranged into 5 equal rows or 7 equal rows without chairs left over. Solution: Check if 53 can be divided into 5 equal rows. 53 divided by 5 is 10 remainder 3 (5 x 10 = 50, 53 - 50 = 3). Step 2: Check if 53 can be divided into 7 equal rows.
   Full step-by-step solution

   Step 1: Check if 53 can be divided into 5 equal rows. 53 divided by 5 is 10 remainder 3 (5 x 10 = 50, 53 - 50 = 3). So there would be 3 chairs left over. Step 2: Check if 53 can be divided into 7 equal rows. 53 divided by 7 is 7 remainder 4 (7 x 7 = 49, 53 - 49 = 4). So there would be 4 chairs left over. Step 3: Now we need to find all the factors of 53. The only numbers that divide 53 evenly are 1 and 53 (1 x 53 = 53). There are no other whole number pairs that multiply to 53. Step 4: Since 53 has exactly two factors (1 and itself), it is a prime number. This means Kaia cannot arrange the chairs into any equal rows other than 1 row or 53 rows. The answer is: 53 is a prime number. It cannot be arranged into 5 equal rows or 7 equal rows without chairs left over.

5. Mere is designing a number chart on a grid. She colors all the composite numbers from 2 to 50 in blue. She then draws a red square around every prime number from 2 to 50. How many numbers from 2 to 50 will have a red square around them? Answer: 15 Solution: Understand that prime numbers are numbers greater than 1 that have exactly two factors (1 and itself).
   Full step-by-step solution

   Step 1: Understand that prime numbers are numbers greater than 1 that have exactly two factors (1 and itself). Step 2: List all numbers from 2 to 50 and identify primes: - 2 (factors: 1, 2) - prime - 3 (factors: 1, 3) - prime - 4 (factors: 1, 2, 4) - composite - 5 (factors: 1, 5) - prime - 6 (factors: 1, 2, 3, 6) - composite - 7 (factors: 1, 7) - prime - 8, 9, 10 - composite - 11 (factors: 1, 11) - prime - 12, 13, 14, 15, 16 - 13 is prime, rest composite - 17 - prime - 18, 19, 20 - 19 is prime, rest composite - 21, 22, 23 - 23 is prime, rest composite - 24, 25, 26, 27, 28, 29 - 29 is prime - 30, 31 - 31 is prime - 32, 33, 34, 35, 36, 37 - 37 is prime - 38, 39, 40, 41 - 41 is prime - 42, 43 - 43 is prime - 44, 45, 46, 47 - 47 is prime - 48, 49, 50 - all composite Step 3: Count the prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. That is 15 numbers. The answer is 15.

6. Emma is organizing her rock collection into display cases. She has 121 rocks and wants to arrange them into equal rows. If she tries to make 11 equal rows, will she have any rocks left over? Explain how you know whether 121 is a prime or composite number. Answer: No, because 121 is a composite number (11 × 11 = 121) Solution: To determine if a number is prime or composite, we check if it has factors other than 1 and itself. The number 17 is prime because its only factors are 1 and 17.
   Full step-by-step solution

   To determine if a number is prime or composite, we check if it has factors other than 1 and itself. For example, the number 15 is composite because it can be divided evenly by 3 and 5 (3 × 5 = 15). The number 17 is prime because its only factors are 1 and 17.