Free Math Worksheets

- Some numbers look prime but have less obvious factors. - Test whether \(77\) is divisible by \(7\). - Calculate the digit sum of \(93\).

1. The number \(77\) is divisible by \(7\), since \(77=7\times11\). Therefore, it is composite. 2. The digit sum of \(93\) is \(9+3=12\), so \(93\) is divisible by \(3\). Since \(93=3\times31\), it is composite.

Neither number is prime. The factorizations are \(77=7\times11\) and \(93=3\times31\).

- How many positive factors does a prime number have? - Which even number is prime? - Test the second statement with small prime numbers.

1. Statement a is true. The number \(2\) is prime and even. Every other even number is divisible by \(2\) and has more than two positive factors. 2. Statement b is false. For example, \(2\times2=4\), which is composite. More generally, twice any prime number has both \(2\) and that prime as factors.

a) True b) False

- A prime number has exactly two positive factors. - Use the digit-sum rule to test divisibility by \(3\). - Check whether each number appears in familiar multiplication facts. - For these numbers, testing small factors is enough.

1. Check possible factor pairs for \(43\). None of \(2\), \(3\), \(4\), \(5\), or \(6\) divides \(43\). Any factor pair using a factor of \(7\) or more would pair it with a smaller factor already tested. Therefore, \(43\) is prime. 2. The digit sum of \(57\) is \(12\), so \(57\) is divisible by \(3\). Since \(57=3\times19\), it is composite. 3. Check possible factor pairs for \(61\). None of \(2\), \(3\), \(4\), \(5\), \(6\), or \(7\) divides \(61\). Any factor pair using a factor of \(8\) or more would pair it with a smaller factor already tested. Therefore, \(61\) is prime. 4. Since \(81=9\times9\), the number \(81\) is composite.

Prime: \(43\) and \(61\) Composite: \(57=3\times19\) and \(81=9\times9\)

- How many positive factors does a prime number have? - Test odd numbers using small factors. - A number may look prime, so verify it carefully.

1. The digit sum of \(87\) is \(8+7=15\), so \(87\) is divisible by \(3\). Since \(87=3\times29\), it is composite. 2. Since \(91=7\times13\), the number \(91\) is composite. 3. Check possible factors of \(97\) from \(2\) through \(9\). None divides \(97\). Any nontrivial factor pair would have a smaller factor in that range, so \(97\) is prime.

\(87\) is composite because \(87=3\times29\). \(91\) is composite because \(91=7\times13\). \(97\) is prime.

- First list all odd numbers in the interval. - Eliminate numbers that appear in familiar multiplication facts. - Other than \(2\) and \(5\), which ones digits can prime numbers have?

1. The odd numbers in the interval are \(21,23,25,27,\) and \(29\). 2. The numbers \(21=3\times7\), \(25=5\times5\), and \(27=3\times9\) are composite. 3. The numbers \(23\) and \(29\) have no positive factors other than \(1\) and themselves, so they are prime.

\(23\) and \(29\)

- Use the digit sum to test divisibility by \(3\). - Check whether a number can be written as a product of familiar factors. - To prove a number is prime, test possible small factors systematically.

1. The digit sum of \(69\) is \(6+9=15\), so \(69\) is divisible by \(3\). Since \(69=3\times23\), it is composite. 2. Since \(77=7\times11\), the number \(77\) is composite. 3. Check possible factors of \(83\) from \(2\) through \(9\). None divides \(83\). Any nontrivial factor pair would have a smaller factor in that range, so \(83\) is prime.

a) \(69\) is composite. b) \(77\) is composite. c) \(83\) is prime.

- Recall the definition of a prime number. - Use divisibility rules where they apply. - Test possible small factors systematically. - Check whether \(77\) is divisible by \(7\).

1. The digit sum of \(39\) is \(12\), so it is divisible by \(3\). Since \(39=3\times13\), it is composite. 2. Check possible factors of \(71\) from \(2\) through \(8\). None divides \(71\). Any nontrivial factor pair would have a smaller factor in that range, so \(71\) is prime. 3. Since \(77=7\times11\), the number \(77\) is composite.

Only \(71\) is prime. The composite numbers can be written as \(39=3\times13\) and \(77=7\times11\).

- What distinguishes a prime number from a composite number? - Use the divisibility rules for \(2\), \(3\), and \(5\). - How can a digit sum test divisibility by \(3\)? - Can an even number greater than \(2\) be prime?

1. The digit sum of \(51\) is \(6\), so \(51\) is divisible by \(3\). Since \(51=3\times17\), it is composite. 2. The number \(52\) is even, so it is divisible by \(2\). Since \(52=2\times26\), it is composite. 3. Check possible factors of \(53\) from \(2\) through \(7\). None divides \(53\). Any nontrivial factor pair would have a smaller factor in that range, so \(53\) is prime.

Only \(53\) is prime. The other numbers are composite because \(51=3\times17\) and \(52=2\times26\).

- Test each number using familiar divisibility rules. - The ones digit may quickly identify a factor. - Use the digit sum to test divisibility by \(3\). - Once you find a factor other than \(1\) and the number itself, the number is composite.

1. The digit sum of \(63\) is \(9\), so \(63\) is divisible by \(3\). In fact, \(63=3\times21\). 2. The number \(65\) ends in \(5\), so it is divisible by \(5\). In fact, \(65=5\times13\). 3. The digit sum of \(69\) is \(15\), so \(69\) is divisible by \(3\). In fact, \(69=3\times23\). 4. Check possible factors of \(67\) from \(2\) through \(8\). None divides \(67\). Any nontrivial factor pair would have a smaller factor in that range, so \(67\) is prime.

The prime number is \(67\). The others are composite because \(63=3\times21\), \(65=5\times13\), and \(69=3\times23\).

- Both numbers in the pair must be prime. - Twin primes differ by exactly \(2\). - Use divisibility rules to test the numbers. - Recall that a prime has exactly two positive factors.

1. The numbers \(5\) and \(7\) are both prime and differ by \(2\), so they are twin primes. 2. The number \(13\) is prime, but \(15=3\times5\) is composite. Therefore, they are not twin primes. 3. The numbers \(29\) and \(31\) are both prime and differ by \(2\), so they are twin primes. 4. The number \(37\) is prime, but \(39=3\times13\) is composite. Therefore, they are not twin primes.

a) Yes b) No; \(15=3\times5\) is composite. c) Yes d) No; \(39=3\times13\) is composite.

- Which two whole numbers are immediately before and after each given even number? - Are both neighboring numbers prime? - Can a divisibility rule quickly show that \(21\) is composite?

1. The neighbors of \(12\) are \(11\) and \(13\). Each has exactly two positive factors, so \((11, 13)\) is a twin-prime pair. 2. The neighbors of \(20\) are \(19\) and \(21\). The number \(19\) is prime, but \(21=3\times7\) is composite, so no twin-prime pair has midpoint \(20\). 3. The neighbors of \(42\) are \(41\) and \(43\). Neither number has a factor from \(2\) through \(6\), so both are prime. Therefore, \((41, 43)\) is a twin-prime pair.

For \(12\): \((11, 13)\) For \(20\): no pair, because \(21\) is composite. For \(42\): \((41, 43)\)

- How can you find the midpoint of two numbers from their sum? - Twin primes are one less and one greater than their midpoint. - Verify that both resulting numbers are prime.

1. The midpoint of two numbers whose sum is \(36\) is \(36\div2=18\). 2. The whole numbers one less and one greater than \(18\) are \(17\) and \(19\). Each has exactly two positive factors, so they form the required twin-prime pair. 3. For a sum of \(28\), the midpoint would be \(28\div2=14\). 4. The neighbors of \(14\) are \(13\) and \(15\). Since \(15=3\times5\) is composite, no twin-prime pair has sum \(28\).

The pair with sum \(36\) is \(17\) and \(19\). There is no twin-prime pair with sum \(28\).

- Test possible factors from \(2\) through \(12\). - A complete rectangle cannot have an incomplete final row.

1. A complete rectangle requires the number of dots in each row to be a factor of \(37\). 2. The only factors of \(37\) are \(1\) and \(37\), so none of the numbers from \(2\) through \(12\) divide \(37\) evenly. 3. In the shown attempt, \(37 \div 6 = 6\) remainder \(1\), leaving one dot outside the complete rows.

No arrangement works because \(37\) is not divisible by any whole number from \(2\) through \(12\).

- How many positive factors does a prime number have? - Check the odd numbers \(3,5,7,9,\ldots\). - Is there an even prime number? - What happens when that prime is added to an odd prime?

1. The number \(9\) is odd but composite because \(9=3\times3\). Therefore, statement a is false. 2. The number \(2\) is the only even prime. Using it with the odd prime \(3\) gives \(2+3=5\), which is odd. Therefore, statement b is false.

a) One counterexample is \(9\). b) One counterexample is \(2+3=5\).