- Which numbers divide each given number without leaving a remainder? - How can the number of factors help you decide between prime and composite?

1. The only positive factors of \(17\) are \(1\) and \(17\), so \(17\) is prime. 2. The number \(21\) has factors besides \(1\) and \(21\), since \(3\times7=21\). Therefore, \(21\) is composite.

a) \(17\) is prime because its only positive factors are \(1\) and \(17\). b) \(21\) is composite because \(3\times7=21\), so it has more than two positive factors.

- Count the positive factors of each number. - A prime number has exactly two positive factors. - A composite number has more than two positive factors.

1. The number \(1\) has only one positive factor, \(1\), so it is neither prime nor composite. 2. The number \(2\) has exactly two positive factors, \(1\) and \(2\), so it is prime. 3. The number \(9\) has positive factors \(1,3,9\), so it is composite.

\(1\): neither prime nor composite because it has only one positive factor, \(1\). \(2\): prime because its only positive factors are \(1\) and \(2\). \(9\): composite because \(9=3\times3\), so it has more than two positive factors.

- Count the rows and the squares in a row. - Use those counts to write a multiplication equation. - Does the total have a factor pair other than one and itself?

1. The rectangle has \(2\) rows of \(3\) squares, so \(2 \times 3 = 6\). 2. Its factors include \(1, 2, 3, 6\). Since \(6\) has more than two factors, it is composite.

\(6\) squares; composite, because \(2 \times 3 = 6\).

- List the positive whole numbers that divide \(1\) exactly. - How many factors must a prime number have? - Does this number meet the rule for a prime or a composite number?

1. The only positive factor of \(1\) is \(1\) itself. 2. A prime number has exactly two positive factors, and a composite number has more than two. Therefore \(1\) is neither.

Neither. \(1\) has only one positive factor.

- What makes a number prime: being odd or having exactly two positive factors? - Can the number be arranged in equal rows beyond a single row? - Try a multiplication fact involving the same factor twice.

1. Being odd does not mean a number has exactly two factors. 2. Since \(3 \times 3 = 9\), the factors of \(9\) include \(1,3,9\). 3. It has more than two factors, so it is composite.

\(9\) is composite, not prime: the array shows \(3 \times 3 = 9\), so its factors include \(1,3,9\).

- Try the multiplication facts that could give the target number. - Does any smaller positive number other than one divide it exactly? - Compare the number of factors with the definition of prime.

1. The positive factors of \(7\) are \(1\) and \(7\). 2. There are exactly two positive factors, so \(7\) is prime.

Prime; its positive factors are \(1\) and \(7\).

- A number that has any factor other than \(1\) and itself is composite. - Use quick divisibility tests where they apply. - If a divisibility rule does not settle a number, test small possible factors systematically.

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

- Count the dots from the diagrams; the totals are not stated in the text. - Compare the two row lengths in each panel. - Think about which composite whole numbers are possible below \(9\) and what they have in common.

1. Panel a) contains \(7\) dots. Its rows have lengths \(4\) and \(3\), so it cannot form \(2\) equal rows. 2. Every composite positive whole number less than \(9\) is divisible by \(2\): the composite possibilities are \(4,6,8\). Therefore, the failed two-row arrangement is enough to conclude that \(7\) is prime. 3. Panel b) contains \(8\) dots in \(2\) equal rows of \(4\), so \(8=2\times4\) and is composite.

Panel a) has \(7\) dots. The rows are \(4\) and \(3\), so they are not equal. Since every composite whole number below \(9\) is even, this rules out a nontrivial factor pair and \(7\) is prime. Panel b) has \(8\) dots in \(2\) equal rows of \(4\), so \(8=2\times4\) and is composite.

- Does being odd rule out all factor pairs? - Test multiplication facts besides the two-times table. - One nontrivial factor pair is enough to prove a number is composite.

1. A number can be odd and still have factors other than \(1\) and itself. 2. Since \(3 \times 7 = 21\), it has factors \(3\) and \(7\) as well as \(1\) and \(21\). 3. It is composite.

\(21\) is composite. The factor pair \((3,7)\) shows why checking only \(2\) is not enough.

- A prime number has exactly two positive factors. - Test small possible factor pairs rather than relying on odd or even alone. - Finding one additional factor pair proves a number is composite.

1. \(19\) has no positive factors other than \(1\) and \(19\), so it is prime. 2. \(21 = 3 \times 7\), so \(21\) is composite. 3. \(23\) has no positive factors other than \(1\) and \(23\), so it is prime.

\(19\) and \(23\) are prime: neither has a factor of \(2\), \(3\), or \(4\), and two factors of \(5\) or more would make at least \(25\). \(21\) is composite because \(3 \times 7 = 21\).

- Does a factor stop being a factor when it appears twice in a multiplication fact? - Write down all the positive factors you know for the number. - Use the definition of prime to check the claim.

1. Multiplying \(7\) by itself still shows that \(7\) is a factor of \(49\). 2. The positive factors include \(1,7,49\), more than the two factors of a prime number. 3. Therefore \(49\) is composite.

Composite. Even when repeated, \(7\) is a factor because \(7 \times 7 = 49\). The number has at least three factors: \(1,7,49\).

- 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 that a prime number has exactly two positive factors. - Use familiar divisibility rules first, then test other small factors systematically. - For a two-digit candidate, you do not need to test factors larger than the smaller member of any possible factor pair.

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 \(71\), 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 \(52=2\times26\).

- Classify every number independently; do not assume how many primes are in the list. - Use familiar divisibility tests to find nontrivial factors quickly. - For a possible prime, check all possible smaller factors needed to rule out a factor pair.

1. The digit sum of \(63\) is \(9\), so \(63=3 \times 21\) and is composite. 2. The number \(65\) ends in \(5\), so \(65=5 \times 13\) and is composite. 3. Test possible smaller factors for \(67\). It is not divisible by \(2,3,5,\) or \(7\); since \(8 \times 8>67\), no nontrivial factor pair remains. Therefore, \(67\) is prime. 4. The digit sum of \(69\) is \(15\), so \(69=3 \times 23\) and is composite.

\(63\): composite because \(63=3\times21\). \(65\): composite because \(65=5\times13\). \(67\): prime because it is not divisible by \(2,3,5,\) or \(7\), and \(8\times8>67\), so no further smaller factor needs to be checked. \(69\): composite because \(69=3\times23\).

- 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 \(28\).

- A complete rectangle needs a row size that is a factor of \(37\). - Use the pictured attempt as one example, then decide whether any other allowed row size works. - A row size works only if there is no incomplete final row.

1. A complete rectangle requires the number of dots in each row to be a factor of \(37\). 2. The only positive factors of \(37\) are \(1\) and \(37\), so none of the numbers from \(2\) through \(12\) works. 3. The diagram uses \(6\) dots in each full row. Since \(37=6\times6+1\), one dot is left over.

No arrangement works because \(37\) has no factor from \(2\) through \(12\).

- Count each total carefully before testing factor pairs. - Try the smaller possible row counts, even if they are not pictured. - What must be true about every possible pair to call a number prime?

1. Group a) has \(11\) dots. Groups of \(2\) or \(3\) leave extras. Two factors both at least \(4\) would make at least \(16\), so its only pair is \((1,11)\). It is prime. 2. Group b) has \(12\) dots. Its factor pairs are \((1,12)\), \((2,6)\), and \((3,4)\). It is composite.

a) \(11\): \((1,11)\), prime. b) \(12\): \((1,12)\), \((2,6)\), \((3,4)\), composite.

- Test each statement with several small cases. - For part a, compare odd numbers with the definition of a prime number. - For part b, think about the possible parity of prime numbers. - Remember that one valid counterexample is enough to disprove a universal statement.

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 \(2+3=5\).

- A factor pair other than one and the number itself proves it is composite. - How large could the smaller factor in a factor pair of a number below \(100\) be? - Check the possible small divisors before deciding that a number is prime.

1. \(91=7\times13\), so \(91\) has a factor pair other than \(1\times91\) and is composite. 2. To test \(97\), consider any factor pair other than \(1\times97\). Its smaller factor would be at most \(9\), since \(10\times10=100\) is already bigger than \(97\). 3. None of \(2,3,4,5,6,7,8,9\) divides \(97\) evenly. So its only factor pair is \(1\times97\), and \(97\) is prime.

a) \(91\) is composite: \(91=7\times13\). b) \(97\) is prime: no whole number from \(2\) through \(9\) divides it evenly, and two factors of at least \(10\) would multiply to at least \(100\).