Number Theory

Problem 701

Determine if the following numbers are prime or composite: 81, 43, 72, 93, 53, 87, 13, 27, 88, 19, 69, 79.

See Solution

Problem 702

Express 52 as a product of its prime factors.

See Solution

Problem 703

Find the prime factors of 70 in ascending order. Recall that 1 is not prime. $[?] \times [] \times []$

See Solution

Problem 704

Find the HCF and LCM of the prime numbers 7 and 11. What are the values of $HCF(7, 11)$ and $LCM(7, 11)$ ?

See Solution

Problem 705

Find the prime factorization of 56.

See Solution

Problem 706

Find the prime factors of 27 and 90. Then calculate the LCM and GCF.
Prime factors: $27=$ $90=$
LCM =

See Solution

Problem 707

Find the prime factors of 42 and 68. Then calculate the LCM and GCF. Prime factors: $42=$ , $68=$ . LCM=$$.

See Solution

Problem 708

What does it mean for an integer $a$ to divide another integer $b$ ? Check your lecture notes for the definition.

See Solution

Problem 709

Find the prime factorization of these numbers: 250 and 36.

See Solution

Problem 710

Find the GCF of 90 and 135.

See Solution

Problem 711

Trouve le nombre de diviseurs pour les nombres $n$ : 20, 34, 54, 70, 86. Remplis les valeurs manquantes $?$ dans la liste.

See Solution

Problem 712

What is the probability of spinning a multiple of 3 and 2 on a 14-section spinner?

See Solution

Problem 713

My number is a multiple of 6. What three other numbers must it also be a multiple of?

See Solution

Problem 714

Factor 165 into its prime components.

See Solution

Problem 715

Find the HCF and LCM of 315 ( $3^2 \times 5 \times 7$ ) and 693 ( $3^2 \times 7 \times 11$ ).

See Solution

Problem 716

Find the LCM and HCF of $\mathrm{G}=2^{5} \times 3^{3} \times 5$ and $\mathrm{H}=2^{2} \times 3^{6} \times 5 \times 7$ .

See Solution

Problem 717

Find the greatest length in meters that Samuel can cut from 99 m of yellow and 165 m of purple ribbon with no leftovers.

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Problem 718

Find the highest common factor (HCF) of 1960 and 7644, given their prime factorizations: $1960=2^{3} \times 5 \times 7^{2}$ and $7644=2^{2} \times 3 \times 7^{2} \times 13$ .

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Problem 719

True or False: 1) a) $x+3=0$ has a solution in $\mathbb{N}$ . b) $\frac{2}{3}$ is a decimal. c) $\sqrt{1.44} \in \mathbb{Z}$ . 2) a) $2x=-1$ has a solution in $\mathbb{Q}^{*}$ . b) $\frac{25}{2}$ is irrational. c) $\pi=3.1416 \ldots$ is a decimal. 3) a) $5x^{2}-25=0$ has solutions in $\mathbb{Q}$ . b) $\sqrt{169}$ is irrational. c) $2.41111 \ldots$ is a decimal. 4) a) $4x^{2}-36=0$ has solutions in $\mathbb{Z}$ . b) $\sqrt{17}$ is irrational. c) $1.555 \ldots$ is rational.

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Problem 720

Find the sum of four different odd integers between 2 and 20: 3 multiples of 3 and 2 multiples of 5.

See Solution

Problem 721

Find common primes in the factor trees for 30 and 42, and determine their HCF as a product of primes.

See Solution

Problem 722

Find the greatest common factor (GCF) of 36 and 3.

See Solution

Problem 723

Nenne 3 Beispiele für "unvernünftige" Zahlen. Für welches $a$ ist $\sqrt{a}$ "vernünftig" oder "unvernünftig"?

See Solution

Problem 724

Create a prime factor tree for 220 and express its prime decomposition as $2^2 \times 5^1 \times 11^1$ .

See Solution

Problem 725

Find the highest common factor (HCF) of 5096 and 6468, given their prime factorizations: $5096=2^{3} \times 7^{2} \times 13$ and $6468=2^{2} \times 3 \times 7^{2} \times 11$ .

See Solution

Problem 726

Find two primes that sum to 30 and then choose an even number greater than 50 to find two primes that sum to it.

See Solution

Problem 727

Find all twin primes less than 100, where twin primes are pairs of consecutive odd primes like 11 and 13.

See Solution

Problem 728

Find all Mersenne primes less than 100, where a Mersenne prime is of the form $2^{n}-1$ .

See Solution

Problem 729

Find all factors of 32 for how many cupcakes Linda can put in each container. Options: A) $1,2,4$ B) $1,2,16,32$ C) $1,2,4,8,16,32$ D) $4,8,16,32$

See Solution

Problem 730

Find the highest common factor (HCF) of 70 and 385 using their prime factors: 2, 5, 7 for both.

See Solution

Problem 731

Find the highest common factor (HCF) of 42 and 231 using their prime factor trees.

See Solution

Problem 732

Find the greatest common factor (GCF) of 30 and 75.

See Solution

Problem 733

Determine if the statement "If a number is divisible by 3, then it is odd" is true or false.

See Solution

Problem 734

Find the prime factors of 72.

See Solution

Problem 735

Find the prime factors of 81.

See Solution

Problem 736

Find the prime factors of 48.

See Solution

Problem 737

Determine the prime factors of 104.

See Solution

Problem 738

Find the prime factors of 66.

See Solution

Problem 739

Find the prime factors of 120.

See Solution

Problem 740

Which statements are equivalent to $N$ is divisible by 160? a, b, c, d, e, f, g, h.

See Solution

Problem 741

Find the LCM and HCF of $E=2^{3} \times 3^{6} \times 5$ and $F=2^{4} \times 3^{2} \times 5 \times 7$ .

See Solution

Problem 742

Find the LCM and HCF of $\mathrm{C}=2^{8} \times 3^{4} \times 5$ and $\mathrm{D}=2^{4} \times 3^{3} \times 5 \times 7$ .

See Solution

Problem 743

Find the greatest length in metres that Samuel can cut from 99 m of yellow and 165 m of purple ribbon without leftovers.

See Solution

Problem 744

Find the highest common factor (HCF) of 1960 and 6468, given their prime factorizations: $1960=2^{3} \times 5 \times 7^{2}$ and $6468=2^{2} \times 3 \times 7^{2} \times 11$ .

See Solution

Problem 745

Chloe finds the prime decomposition of 208 as $2^{a} \times b$ . What are the values of $a$ and $b$ ?

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Problem 746

Classify each number as prime or composite: 88, 23, 39, 51, 61, 67, 41, 99, 201, 87. Explain your reasoning.

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Problem 747

Find the prime factors of 63. The prime factorization of 63 is $\square$ .

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Problem 748

Find pairs of numbers where the least common multiple equals their product. Also, determine when two signs blink together again.

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Problem 749

Find the prime factors of 18: the answer is $\square$ .

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Problem 750

Find the prime factors of the composite number 9. a) $1 \times 9$ b) $3 \times 3$

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Problem 751

Draw the prime factor tree for 5 and find the LCM of 50 and 525.

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Problem 752

Find the greatest common factor (GCF) of the pairs: 18 and 45, and 56 and 24. GCF of 18 and 45 is $. GCF of 56 and 24 is$ .

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Problem 753

Find the first 5 multiples of 16 and list all factors of 16. Is 16 prime or composite?

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Problem 754

Find the largest divisor of 10985 that is less than 10985.

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Problem 755

Find the common factors of 45 and 60.

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Problem 756

Circle the prime numbers: 21, $13$ , 36, 4, 31, 23, 39, 27. Underline the composite numbers: 45, 11, 30, 47, 71, 99, 81, 67, 53, 63.

See Solution

Problem 757

Find the prime factorization of 45.

See Solution

Problem 758

Factor 70 into its prime components.

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Problem 759

Find the LCM of 12 and 22.

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Problem 760

Find the product of different prime numbers $a, b, c$ if $abc^2 = 140$ . What is $abc$ ? A. 140 B. 120 C. 70 D. 14 E. 16

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Problem 761

Find the prime factorization of 630.

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Problem 762

True or False: The GCF of 39 and 51 is 3.

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Problem 763

Which number is a perfect square: 115, 154, 169, or 279?

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Problem 764

Find the prime factorization of 1,260. Choose from: A. $2 \times 3 \times 5 \times 6 \times 7$ , B. $2 \times 3 \times 5 \times 7$ , C. $4 \times 5 \times 7 \times 9$ , D. $2 \times 2 \times 3 \times 3 \times 5 \times 7$ .

See Solution

Problem 765

Jan has 6, 8, or 12 classmates. What is the least number of pizza slices for equal sharing with no leftovers? A. 12 B. 16 C. 24 D. 32 E. 36

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Problem 766

Identify the composite number from the options: A. 19, B. 1, C. 63, D. 0.

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Problem 767

Which option is a composite number: A. 31, B. 13, C. 63, or D. 61?

See Solution

Problem 768

Which group contains only prime numbers? A. $2,5,15,19$ B. $7,17,29,49$ C. $2,3,5,9$ D. $3,11,23,31$

See Solution

Problem 769

Find all in-between primes less than 100, where a number $n$ is in-between if both $n-1$ and $n+1$ are prime.

See Solution

Problem 770

Find the prime factorization and perfect squares for: a) 42, b) 169, c) 256.

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Problem 771

Add $232_{\text{five}}$ and $34_{\text{five}}$ . Find total flats, longs, units, and regroup as indicated. Result: $232_{\text{five}} + 34_{\text{five}} = \square$ .

See Solution

Problem 772

Find the prime factorization of 61. If it's prime, just write 61. Answer: $61 =$

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Problem 773

Identify the common prime factors of 30 and 42, then find and calculate their highest common factor (HCF).

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Problem 774

14 is factored into primes. a) Complete the prime factor tree for 21 and write it as a product of primes. b) Find the LCM of 14 and 21.

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Problem 775

Find the HCF of 42 and 231 using prime factorization: 42 is $2 \times 3 \times 7$ and 231 is $3 \times 7 \times 11$ .

See Solution

Problem 776

Complete the prime factor trees for 21 and 77. Find the lowest common multiple (LCM) of 21 and 77.

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Problem 777

Find the LCM of 42 and 165 using their prime factors: 42 = $2 \times 3 \times 7$ , 165 = $3 \times 5 \times 11$ .

See Solution

Problem 778

حلل العدد $C=18 \times 5^{n}$ إلى عوامل أولية حيث $n$ عدد طبيعي و $n<4$ .

See Solution

Problem 779

Find the divisors of $2^{5} \cdot 3^{5} \cdot 11^{7}$ that are multiples of 12 and those that are not.

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Problem 780

Find the greatest common factor of 4, 10, and 16, then list them from least to greatest.

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Problem 781

Find the prime factorization of $24$ .

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Problem 782

Find the check digit for the credit card number 516281474621 911d using the digits 6, 7, 8, and 9.

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Problem 783

Determine the base of the equation $!!\&\& \% !!!\&\&\&\&\&\& = *!!!!!\&\&\&\&$ and convert it to base 10.

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Problem 784

Identify two numbers from the Venn diagram: left circle has 3, right has 2, 2; intersection has 5. Find GCF and LCM.

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Problem 785

Find the prime factors of 90.

See Solution

Problem 786

Find the greatest common factor of 45, 135, and 250. A. 3 B. 5 C. 15 D. 25 E. 45

See Solution

Problem 787

Ester needs a number between 1 and 40 that divides 297. What is the probability of choosing a valid divisor?

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Problem 788

Which expression must be composite if $x$ is a prime number? A. $x^{2}+1$ B. $x^{2}+2$ C. $x^{2}+3$ D. $x^{2}+4$ E. $x^{2}+5$

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Problem 789

a) Factor 98 into prime factors in ascending order. b) Find the HCF of 98 and 42.

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Problem 790

Find the prime factorization for these numbers: 90, 66, 52, and 41 (if prime, state "prime").

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Problem 791

Determine if 3,600 is a perfect square and find its square root.

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Problem 792

Aufgaben: 1 a) Ist 35 ein Teiler von $70+355$ ? b) Ist 8 ein Teiler von $48+800+160$ ? c) Ist 25 ein Teiler von $7800-75$ ? d) Ist 7 ein Teiler von $420-35-12$ ? 2 Prüfe, ob 38, 45, 31, 25 die angegebenen Summen/Differenzen teilen. 3 Prüfe die Teilbarkeit der Summen durch 2 oder 5. 4 Ist 23050 durch 20, und ist 47920 durch 40 teilbar?

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Problem 793

What is the smallest number with exactly four different prime factors?

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Problem 794

Determine if the following numbers are prime: a) 18 b) 11 c) 1

See Solution

Problem 795

Identify the prime numbers in the list: 16, 1, 13, 20, 19, 15.

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Problem 796

Find the largest 10-digit number using each digit 0-9 once, that is a multiple of 36.

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Problem 797

Draw the prime factor tree for 210 and find the highest common factor (HCF) of 210 and 693.

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Problem 798

Create a prime factor tree for 126 and express its prime decomposition in index form as $2^1 \times 3^2 \times 7^1$ .

See Solution

Problem 799

Find the lowest common multiple (LCM) of 70 and 273 using prime factor trees.

See Solution

Problem 800

Draw the prime factor tree for 190 and find the LCM of 105 and 190.

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Number Theory

Problem 701

Determine if the following numbers are prime or composite: 81, 43, 72, 93, 53, 87, 13, 27, 88, 19, 69, 79.

Problem 702

Express 52 as a product of its prime factors.

Problem 703

Find the prime factors of 70 in ascending order. Recall that 1 is not prime. [?]×[]×[] [?] \times [] \times [] [?]×[]×[]

Problem 704

Find the HCF and LCM of the prime numbers 7 and 11. What are the values of HCF(7,11)HCF(7, 11)HCF(7,11) and LCM(7,11)LCM(7, 11)LCM(7,11)?

Problem 705

Find the prime factorization of 56.

Problem 706

Find the prime factors of 27 and 90. Then calculate the LCM and GCF. Prime factors: 27=27=27= 90=90=90= LCM =

Problem 707

Find the prime factors of 42 and 68. Then calculate the LCM and GCF. Prime factors: 42=42=42=, 68=68=68=. LCM=$$.

Problem 708

What does it mean for an integer aaa to divide another integer bbb? Check your lecture notes for the definition.

Problem 709

Find the prime factorization of these numbers: 250 and 36.

Problem 710

Find the GCF of 90 and 135.

Problem 711

Trouve le nombre de diviseurs pour les nombres nnn: 20, 34, 54, 70, 86. Remplis les valeurs manquantes ??? dans la liste.

Problem 712

What is the probability of spinning a multiple of 3 and 2 on a 14-section spinner?

Problem 713

My number is a multiple of 6. What three other numbers must it also be a multiple of?

Problem 714

Factor 165 into its prime components.

Problem 715

Find the HCF and LCM of 315 (32×5×73^2 \times 5 \times 732×5×7) and 693 (32×7×113^2 \times 7 \times 1132×7×11).

Problem 716

Find the LCM and HCF of G=25×33×5\mathrm{G}=2^{5} \times 3^{3} \times 5G=25×33×5 and H=22×36×5×7\mathrm{H}=2^{2} \times 3^{6} \times 5 \times 7H=22×36×5×7.

Problem 717

Find the greatest length in meters that Samuel can cut from 99 m of yellow and 165 m of purple ribbon with no leftovers.

Problem 718

Find the highest common factor (HCF) of 1960 and 7644, given their prime factorizations: 1960=23×5×721960=2^{3} \times 5 \times 7^{2}1960=23×5×72 and 7644=22×3×72×137644=2^{2} \times 3 \times 7^{2} \times 137644=22×3×72×13.

Problem 719

Problem 720

Find the sum of four different odd integers between 2 and 20: 3 multiples of 3 and 2 multiples of 5.

Problem 721

Find common primes in the factor trees for 30 and 42, and determine their HCF as a product of primes.

Problem 722

Find the greatest common factor (GCF) of 36 and 3.

Problem 723

Nenne 3 Beispiele für "unvernünftige" Zahlen. Für welches aaa ist a\sqrt{a}a​ "vernünftig" oder "unvernünftig"?

Problem 724

Create a prime factor tree for 220 and express its prime decomposition as 22×51×1112^2 \times 5^1 \times 11^122×51×111.

Problem 725

Find the highest common factor (HCF) of 5096 and 6468, given their prime factorizations: 5096=23×72×135096=2^{3} \times 7^{2} \times 135096=23×72×13 and 6468=22×3×72×116468=2^{2} \times 3 \times 7^{2} \times 116468=22×3×72×11.

Problem 726

Find two primes that sum to 30 and then choose an even number greater than 50 to find two primes that sum to it.

Problem 727

Find all twin primes less than 100, where twin primes are pairs of consecutive odd primes like 11 and 13.

Problem 728

Find all Mersenne primes less than 100, where a Mersenne prime is of the form 2n−12^{n}-12n−1.

Problem 729

Find all factors of 32 for how many cupcakes Linda can put in each container. Options: A) 1,2,41,2,41,2,4 B) 1,2,16,321,2,16,321,2,16,32 C) 1,2,4,8,16,321,2,4,8,16,321,2,4,8,16,32 D) 4,8,16,324,8,16,324,8,16,32

Problem 730

Find the highest common factor (HCF) of 70 and 385 using their prime factors: 2, 5, 7 for both.

Problem 731

Find the highest common factor (HCF) of 42 and 231 using their prime factor trees.

Problem 732

Find the greatest common factor (GCF) of 30 and 75.

Problem 733

Determine if the statement "If a number is divisible by 3, then it is odd" is true or false.

Problem 734

Find the prime factors of 72.

Problem 735

Find the prime factors of 81.

Problem 736

Find the prime factors of 48.

Problem 737

Determine the prime factors of 104.

Problem 738

Find the prime factors of 66.

Problem 739

Find the prime factors of 120.

Problem 740

Which statements are equivalent to NNN is divisible by 160? a, b, c, d, e, f, g, h.

Find the prime factors of 70 in ascending order. Recall that 1 is not prime. $[?] \times [] \times []$

Find the HCF and LCM of the prime numbers 7 and 11. What are the values of $HCF(7, 11)$ and $LCM(7, 11)$ ?

Find the prime factors of 27 and 90. Then calculate the LCM and GCF.
Prime factors: $27=$ $90=$
LCM =

Find the prime factors of 42 and 68. Then calculate the LCM and GCF. Prime factors: $42=$ , $68=$ . LCM=$$.

What does it mean for an integer $a$ to divide another integer $b$ ? Check your lecture notes for the definition.

Trouve le nombre de diviseurs pour les nombres $n$ : 20, 34, 54, 70, 86. Remplis les valeurs manquantes $?$ dans la liste.

Find the HCF and LCM of 315 ( $3^2 \times 5 \times 7$ ) and 693 ( $3^2 \times 7 \times 11$ ).

Find the LCM and HCF of $\mathrm{G}=2^{5} \times 3^{3} \times 5$ and $\mathrm{H}=2^{2} \times 3^{6} \times 5 \times 7$ .

Find the highest common factor (HCF) of 1960 and 7644, given their prime factorizations: $1960=2^{3} \times 5 \times 7^{2}$ and $7644=2^{2} \times 3 \times 7^{2} \times 13$ .

Nenne 3 Beispiele für "unvernünftige" Zahlen. Für welches $a$ ist $\sqrt{a}$ "vernünftig" oder "unvernünftig"?

Create a prime factor tree for 220 and express its prime decomposition as $2^2 \times 5^1 \times 11^1$ .

Find the highest common factor (HCF) of 5096 and 6468, given their prime factorizations: $5096=2^{3} \times 7^{2} \times 13$ and $6468=2^{2} \times 3 \times 7^{2} \times 11$ .

Find all Mersenne primes less than 100, where a Mersenne prime is of the form $2^{n}-1$ .

Find all factors of 32 for how many cupcakes Linda can put in each container. Options: A) $1,2,4$ B) $1,2,16,32$ C) $1,2,4,8,16,32$ D) $4,8,16,32$

Which statements are equivalent to $N$ is divisible by 160? a, b, c, d, e, f, g, h.

Find the LCM and HCF of $E=2^{3} \times 3^{6} \times 5$ and $F=2^{4} \times 3^{2} \times 5 \times 7$ .

Find the LCM and HCF of $\mathrm{C}=2^{8} \times 3^{4} \times 5$ and $\mathrm{D}=2^{4} \times 3^{3} \times 5 \times 7$ .

Find the highest common factor (HCF) of 1960 and 6468, given their prime factorizations: $1960=2^{3} \times 5 \times 7^{2}$ and $6468=2^{2} \times 3 \times 7^{2} \times 11$ .

Chloe finds the prime decomposition of 208 as $2^{a} \times b$ . What are the values of $a$ and $b$ ?

Find the prime factors of 63. The prime factorization of 63 is $\square$ .

Find the prime factors of 18: the answer is $\square$ .

Find the prime factors of the composite number 9. a) $1 \times 9$ b) $3 \times 3$

Find the greatest common factor (GCF) of the pairs: 18 and 45, and 56 and 24. GCF of 18 and 45 is $. GCF of 56 and 24 is$ .

Circle the prime numbers: 21, $13$ , 36, 4, 31, 23, 39, 27. Underline the composite numbers: 45, 11, 30, 47, 71, 99, 81, 67, 53, 63.

Find the product of different prime numbers $a, b, c$ if $abc^2 = 140$ . What is $abc$ ? A. 140 B. 120 C. 70 D. 14 E. 16

Which group contains only prime numbers? A. $2,5,15,19$ B. $7,17,29,49$ C. $2,3,5,9$ D. $3,11,23,31$

Find all in-between primes less than 100, where a number $n$ is in-between if both $n-1$ and $n+1$ are prime.

Add $232_{\text{five}}$ and $34_{\text{five}}$ . Find total flats, longs, units, and regroup as indicated. Result: $232_{\text{five}} + 34_{\text{five}} = \square$ .

Find the prime factorization of 61. If it's prime, just write 61. Answer: $61 =$