Number Theory

Problem 1001

Determine if these statements are true or false, providing a counterexample if false:
1. If $a \mid b$ , then $a \mid(12 \times b)$ .
2. If $a \mid b$ and $a \mid c$ , then $a \mid(b+c)$ .
3. If $a \mid b$ and $a \nmid c$ , then $a \nmid(b+c)$ .

See Solution

Problem 1002

Determine if these statements are true or false. Provide a counterexample if false.
1. If $a \mid b$ , then $a \mid(12 \times b)$ . A. true B. false
2. If $a \mid b$ and $a \mid c$ , then $a \mid(b+c)$ . A. true B. false
3. If $a \mid b$ and $a \nmid c$ , then $a \nmid(b+c)$ . A. true B. false
4. If $a \mid c$ , then $a \mid b c$ . A. true B. false

See Solution

Problem 1003

Find the highest common factor (HCF) of 12600 and 14112 given their prime factorizations:
$12600=2^{3} \times 3^{2} \times 5^{2} \times 7$ and $14112=2^{5} \times 3^{2} \times 7^{2}$ .

See Solution

Problem 1004

Décompose les nombres suivants en produits de facteurs premiers : 150, 300, 270, 700, 36, 200, 228, 145.

See Solution

Problem 1005

Find the sum of all possible values for the number of factors $d$ of $m \times n$ , where $m$ and $n$ each have 6 factors.

See Solution

Problem 1006

Find the sum of all possible values of $d$ , where $d$ is the number of factors of $m \times n$ , given $m$ and $n$ each have 6 factors.

See Solution

Problem 1007

Find the factor pairs of 54 and identify the pair with a specific sum. Use $54 = a \times b$ .

See Solution

Problem 1008

Create prime factor trees for 30 and 130, then determine the highest common factor (HCF) of 30 and 130.

See Solution

Problem 1009

Create a prime factor tree for 330 and find the HCF with 308. Explain both the tree and HCF process.

See Solution

Problem 1010

Match the prime and composite numbers: Prime: 11, 19; Composite: 24, 49.

See Solution

Problem 1011

Find all factors of 99. The factors are: $1, 3, 9, 11, 33, 99$ .

See Solution

Problem 1012

Identify the correct prime factorization of 504 in exponential form: $2^{3} \cdot 3^{2} \cdot 7^{2}$ , $2 \cdot 3 \cdot 7$ , $2 \cdot 3^{2} \cdot 7$ , or $2^{3} \cdot 3^{2} \cdot 7$ .

See Solution

Problem 1013

Find the GCF of 120 and 750 using prime factorization. The GCF is $\square$ .

See Solution

Problem 1014

Factor 30 into prime factors using a factor tree if needed. $30=\square \times \square \times \square$

See Solution

Problem 1015

Find the sum of all possible values for the number of factors $d$ of $m \times n$ , where $m$ and $n$ each have 6 factors.

See Solution

Problem 1016

Find the least common multiple (LCM) of 6 and 8. Options: 12, 24, 36, 48.

See Solution

Problem 1017

Find the GCF of 25 and 35. Choices: 1, 5, 7, 25.

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Problem 1018

Identify which of the following numbers are factors of 16,632, given its prime factorization: $2^{3} \times 3^{3} \times 7 \times 11$ .
1. $81 = 3^{4}$
2. $104 = 2^{3} \times 13$
3. $33 = 3 \times 11$
4. $28 = 2^{2} \times 7$
5. $27 = 3^{3}$

See Solution

Problem 1019

Find the prime factorization of the number 126.

See Solution

Problem 1020

Find the prime factorization of 44. Write it as a product of prime factors: $44 =$

See Solution

Problem 1021

Draw a prime factor tree for 330 and use it to find the HCF of 308 and 330.

See Solution

Problem 1022

Express 36 as a product of its prime factors.

See Solution

Problem 1023

Create a prime factor tree for 170 and find the LCM of 105 and 170.

See Solution

Problem 1024

Write the prime factorization of 72.

See Solution

Problem 1025

Find the HCF of 70 and 385 using prime factorization. Correct the factors: 70 = $2 \times 5 \times 7$ , 385 = $5 \times 7 \times 11$ .

See Solution

Problem 1026

Check if these congruences are true: 1) $4 \equiv 19 \,(\bmod 3)$ , 2) $3 \equiv 20 \,(\bmod 3)$ , 3) $9 \equiv 33 \,(\bmod 4)$ , 4) $5 \equiv 20 \,(\bmod 4)$ , 5) $11 \equiv 30 \,(\bmod 5)$ .

See Solution

Problem 1027

Find two numbers that have exactly 3 factors and list all their factors.

See Solution

Problem 1028

A spinner numbered 1 to 8 is spun. Find the probabilities for these events:
a. Factor of 35. b. Multiple of 4. c. Even. d. 3 or 8. e. 9. f. Composite. g. Neither prime nor composite.

See Solution

Problem 1029

Find $\log (1,000,000)$ without a calculator. $\log (1,000,000)=\square($ Type an integer or a decimal. $)$

See Solution

Problem 1030

Write 30 as a sum of two prime numbers. Choose from: A. $30=28+\square$ , B. $30=19+\square$ , C. $30=2+\square$ .

See Solution

Problem 1031

True or false: There are no even prime numbers. Choose the correct answer: A, B, C, or D.

See Solution

Problem 1032

Is 30,031 prime or composite? If composite, provide its prime factorization. A. Composite, factorization: ____ B. Prime.

See Solution

Problem 1033

Find the smallest Smith number, which has equal digit sum to the sum of its prime factors.

See Solution

Problem 1034

Determine if 7 is a Sophie Germain prime by calculating $2p + 1$ and checking if it's prime. What is $2p + 1 = \square$ ?

See Solution

Problem 1035

Find the number of divisors of $7^{8} \cdot 19^{2}$ . There are $\square$ divisors of this composite number.

See Solution

Problem 1036

Verify if 945 is abundant by checking if the sum of its proper divisors exceeds 945. A. Sum is 975, B. Sum is 785.

See Solution

Problem 1037

Determine if the formula $n! \pm 1$ yields a prime for $n = 2, 3, 4, 5$ . Does it produce primes never, sometimes, or always?

See Solution

Problem 1038

Show that 1184 and 1210 are amicable by using their proper divisors. Which statement is correct? A, B, or C?

See Solution

Problem 1039

Find the sixth Fermat number: $2^{2^{5}}+1$ . Then, divide it by 641. What are the results?

See Solution

Problem 1040

Is $35,390,008$ a happy number? Yes/No Is $39,259,592$ a happy number? Yes/No Are $35,390,008$ and $39,259,592$ a happy amicable pair? Yes/No

See Solution

Problem 1041

Find the next two primes of the form $4k+1$ after $5, 13, 17, 29$ and express them as sums of squares.

See Solution

Problem 1042

Find the GCF and LCM of $2^{32} \cdot 5^{17} \cdot 7^{22}$ and $2^{35} \cdot 5^{22} \cdot 7^{14}$ .

See Solution

Problem 1043

Find the LCM of 30, 70, and 80 using prime factors. Provide factorizations and the simplified LCM.

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Problem 1044

Find the prime numbers between 100 and $200$ . How many are there? Choices: A. 24 B. 23 C. 21 D. 19

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Problem 1045

Verify if 836 is a weird number by finding its prime factorization, divisors, and checking if it's abundant.

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Problem 1046

Calculate the difference between the sum of the largest three primes under 20 and the product of the smallest three primes.

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Problem 1047

What number has three different prime factors and two identical digits?

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Problem 1048

I am a two-digit prime number, one less than a multiple of 10. Adding 30 gives another prime. What am I?

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Problem 1049

I am a two-digit prime number, one less than a multiple of 10. Adding 30 gives another prime. What number am I?

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Problem 1050

Which expression correctly represents the prime factorization of 900? a) $2 \times 3^{2} \times 5^{3}$ b) $2^{3} \times 3^{2} \times 5$ c) $3^{2} \times 10^{2}$ d) $2^{2} \times 3^{2} \times 5^{2}$ e) $2^{2} \times 5^{2} \times 9$ f) $2^{2} \times 3 \times 5^{2}$

See Solution

Problem 1051

Find the highest common factor (HCF) of $A = 2^2 \times 3^4 \times 7$ and $B = 3^2 \times 7^2$ .

See Solution

Problem 1052

Find the HCF and LCM of $A = 2^{2} \times 3^{4} \times 7$ and $B = 3^{2} \times 7^{2}$ .

See Solution

Problem 1053

Are the numbers $7,276,766,187$ and 4 relatively prime? Answer: Yes or No.

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Problem 1054

Show that $7,276,766,187$ and 4 are relatively prime by finding the GCD or checking divisibility.

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Problem 1055

List the factors of 30 and find the probability that a randomly chosen factor is a 2-digit number.

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Problem 1056

Explain why the product of two negative reciprocals is always -1.

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Problem 1057

Is the conjecture "All numbers that end in 1 are prime" valid? If not, give a counterexample.

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Problem 1058

Compare the LCM of 4 and 10 with the LCM of 6 and 8. Which is larger? Explain your reasoning.

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Problem 1059

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 1060

Si $p$ est un nombre premier, est-ce que $p^{2}$ est également un nombre premier?

See Solution

Problem 1061

Find the probability of spinning a factor of 15 on an 8-sector spinner numbered 1 to 8. $P($ factor of 15 $)=\square$

See Solution

Problem 1062

Find a value for $a$ such that $a \bmod 5 = 3$ . How many values of $a$ satisfy this condition?

See Solution

Problem 1063

Find a value for $n$ such that $108 \bmod n = 0$ . How many such values for $n$ exist?

See Solution

Problem 1064

Find the greatest common factor (GCF) of 45 and 75. Options: (A) 3 (B) 4 (C) 15 (D) 30.

See Solution

Problem 1065

Find the LCM for each pair: (8, 12), (2, 6), (3, 8), (6, 8), (3, 4). Options: $6, 12, 24, 48$ .

See Solution

Problem 1066

Find the prime factors of 132.

See Solution

Problem 1067

Find the least common multiple (LCM) of 24 and 16 using multiples or prime factors methods.

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Problem 1068

Jason has 22 strawberry and 33 blackberry scones. What’s the max identical bags he can make? Find the GCF of 22 and 33.

See Solution

Problem 1069

Leia's favorite number has factors 2, 5, and 7. Which statements about it are true?

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Problem 1070

Convert 3716 to Roman numerals and solve XXVII = 80.

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Problem 1071

Find the LCM for each pair: (8,12), (2,6), (3,8), (6,8), (3,4) with options 6, 12, 24, 48.

See Solution

Problem 1072

Identify the correct prime factorization of 200 in exponential form: $2^{2} \cdot 5^{2}$ , $2 \cdot 10^{2}$ , $8 \cdot 25$ , or $2^{3} \cdot 5^{2}$ .

See Solution

Problem 1073

Find the prime factorization of 770.

See Solution

Problem 1074

Find the prime factorization of 33.

See Solution

Problem 1075

Find the prime factorization of $156$ .

See Solution

Problem 1076

Find the prime factorization of $126$ .

See Solution

Problem 1077

Find the prime factorization of 750.

See Solution

Problem 1078

Find the prime factorization of 9, expressed as a product of prime numbers.

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Problem 1079

Find the prime factorization of 315.

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Problem 1080

Find the prime factors of $686$ .

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Problem 1081

Find the prime factorization of $98$ .

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Problem 1082

Find the GCF of $2^{4} \times 11^{6}$ and $2^{1} \times 11^{3}$ .

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Problem 1083

Find the prime factorization of: 18, 120, 56, 390, 144, 153. Then, find the GCF of: (16,20), (15,28), (33,66), (78,30), (9,36), (35,42), (100,120), (84,42). Also, find GCF of monomials: (15 x^{4}, 35 x^{2}), (-6 t^{3}, 9 t), (12 a b, 12), (-m^{8} n^{4}, 3 m^{6} n). Lastly, for Kirstin's photos (36 family, 28 friends), find the max rows and photos per row.

See Solution

Problem 1084

Find the prime factorization of 84 using the numbers provided. Use each number as needed. $84 = 2^2 \times 3 \times 7$

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Problem 1085

Decide if each number is prime or composite: 3, 8, 9, 17, 18. Choose Prime or Composite for each.

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Problem 1086

Find the GCF and LCM of 8 and 12.

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Problem 1087

Determine if 12, 10, 9, and 5 are common factors of 36 and 90.

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Problem 1088

Critique Rob's claim that all numbers have an even number of factors. Marcia says some have an odd number. Who is right?

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Problem 1089

Find the factors of 38, 39, and 40. Do they share any common factors? Explain how to determine this without listing factors.

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Problem 1090

Critique Rob's claim that all numbers have even factors. Marcia says some have odd factors. Who is right? Explain.

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Problem 1091

Find the prime factorization of 147 and 259 to simplify the fraction $\frac{147}{259}$ .

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Problem 1092

Find the GCF of 30 and 12, and the LCM of 3 and 6. GCF: $?, LCM:$ ?

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Problem 1093

Fill in the GCF for these pairs: 8 and 12, 12 and 30, 22 and 36.

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Problem 1094

Identify the composite numbers from this list: 49, 35, 32, 41, 47.

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Problem 1095

Identify the true statement: LCM(3,4)=1, LCM(2,10)=10, GCF(7,42)=1, GCF(68,40)=8.

See Solution

Problem 1096

Determine if the products 263, 2205, and 3600 are divisible (Y) or not (N) by 2, 3, 4, 5, 6, 7, 8, 9, 10.

See Solution

Problem 1097

Select two different two-digit numbers from 10 to 30. Calculate their highest common factor (HCF) and lowest common multiple (LCM).

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Problem 1098

Find the GCF of 3 and $12u$ .

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Problem 1099

Find the greatest common factor of these numbers: $2^{4} \cdot 3^{4} \cdot 5^{6} \cdot 7$ , $2^{3} \cdot 3^{6} \cdot 5^{2} \cdot 11$ , $3^{7} \cdot 5^{3} \cdot 7^{2} \cdot 11^{2} \cdot 13$ . Answer as a single number.

See Solution

Problem 1100

Find the following modular calculations: (a) $46 \mod 5$ , (b) $175 \mod 13$ , (c) $(43+29) \mod 17$ , (d) $(14 \times 27) \mod 8$ .

See Solution

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

Number Theory

Problem 1001

Problem 1002

Problem 1003

Find the highest common factor (HCF) of 12600 and 14112 given their prime factorizations: 12600=23×32×52×712600=2^{3} \times 3^{2} \times 5^{2} \times 712600=23×32×52×7 and 14112=25×32×7214112=2^{5} \times 3^{2} \times 7^{2}14112=25×32×72.

Problem 1004

Décompose les nombres suivants en produits de facteurs premiers : 150, 300, 270, 700, 36, 200, 228, 145.

Problem 1005

Find the sum of all possible values for the number of factors ddd of m×nm \times nm×n, where mmm and nnn each have 6 factors.

Problem 1006

Find the sum of all possible values of ddd, where ddd is the number of factors of m×nm \times nm×n, given mmm and nnn each have 6 factors.

Problem 1007

Find the factor pairs of 54 and identify the pair with a specific sum. Use 54=a×b54 = a \times b54=a×b.

Problem 1008

Create prime factor trees for 30 and 130, then determine the highest common factor (HCF) of 30 and 130.

Problem 1009

Create a prime factor tree for 330 and find the HCF with 308. Explain both the tree and HCF process.

Problem 1010

Match the prime and composite numbers: Prime: 11, 19; Composite: 24, 49.

Problem 1011

Find all factors of 99. The factors are: 1,3,9,11,33,991, 3, 9, 11, 33, 991,3,9,11,33,99.

Problem 1012

Identify the correct prime factorization of 504 in exponential form: 23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}23⋅32⋅72, 2⋅3⋅72 \cdot 3 \cdot 72⋅3⋅7, 2⋅32⋅72 \cdot 3^{2} \cdot 72⋅32⋅7, or 23⋅32⋅72^{3} \cdot 3^{2} \cdot 723⋅32⋅7.

Problem 1013

Find the GCF of 120 and 750 using prime factorization. The GCF is □\square□.

Problem 1014

Factor 30 into prime factors using a factor tree if needed. 30=□×□×□ 30=\square \times \square \times \square 30=□×□×□

Problem 1015

Find the sum of all possible values for the number of factors ddd of m×nm \times nm×n, where mmm and nnn each have 6 factors.

Problem 1016

Find the least common multiple (LCM) of 6 and 8. Options: 12, 24, 36, 48.

Problem 1017

Find the GCF of 25 and 35. Choices: 1, 5, 7, 25.

Problem 1018

Problem 1019

Find the prime factorization of the number 126.

Problem 1020

Find the prime factorization of 44. Write it as a product of prime factors: 44= 44 = 44=

Problem 1021

Draw a prime factor tree for 330 and use it to find the HCF of 308 and 330.

Problem 1022

Express 36 as a product of its prime factors.

Problem 1023

Create a prime factor tree for 170 and find the LCM of 105 and 170.

Problem 1024

Write the prime factorization of 72.

Problem 1025

Find the HCF of 70 and 385 using prime factorization. Correct the factors: 70 = 2×5×72 \times 5 \times 72×5×7, 385 = 5×7×115 \times 7 \times 115×7×11.

Problem 1026

Problem 1027

Find two numbers that have exactly 3 factors and list all their factors.

Problem 1028

A spinner numbered 1 to 8 is spun. Find the probabilities for these events: a. Factor of 35. b. Multiple of 4. c. Even. d. 3 or 8. e. 9. f. Composite. g. Neither prime nor composite.

Problem 1029

Find log⁡(1,000,000)\log (1,000,000)log(1,000,000) without a calculator. log⁡(1,000,000)=□(\log (1,000,000)=\square(log(1,000,000)=□( Type an integer or a decimal. )))

Problem 1030

Write 30 as a sum of two prime numbers. Choose from: A. 30=28+□30=28+\square30=28+□, B. 30=19+□30=19+\square30=19+□, C. 30=2+□30=2+\square30=2+□.

Problem 1031

True or false: There are no even prime numbers. Choose the correct answer: A, B, C, or D.

Problem 1032

Is 30,031 prime or composite? If composite, provide its prime factorization. A. Composite, factorization: ____ B. Prime.

Problem 1033

Find the smallest Smith number, which has equal digit sum to the sum of its prime factors.

Problem 1034

Determine if 7 is a Sophie Germain prime by calculating 2p+12p + 12p+1 and checking if it's prime. What is 2p+1=□2p + 1 = \square2p+1=□?

Problem 1035

Find the number of divisors of 78⋅1927^{8} \cdot 19^{2}78⋅192. There are □\square□ divisors of this composite number.

Problem 1036

Verify if 945 is abundant by checking if the sum of its proper divisors exceeds 945. A. Sum is 975, B. Sum is 785.

Problem 1037

Determine if the formula n!±1n! \pm 1n!±1 yields a prime for n=2,3,4,5n = 2, 3, 4, 5n=2,3,4,5. Does it produce primes never, sometimes, or always?

Problem 1038

Show that 1184 and 1210 are amicable by using their proper divisors. Which statement is correct? A, B, or C?

Problem 1039

Find the sixth Fermat number: 225+12^{2^{5}}+1225+1. Then, divide it by 641. What are the results?

Problem 1040

Is 35,390,00835,390,00835,390,008 a happy number? Yes/No Is 39,259,59239,259,59239,259,592 a happy number? Yes/No Are 35,390,00835,390,00835,390,008 and 39,259,59239,259,59239,259,592 a happy amicable pair? Yes/No

Problem 1041

Find the next two primes of the form 4k+14k+14k+1 after 5,13,17,295, 13, 17, 295,13,17,29 and express them as sums of squares.

Problem 1042

Find the highest common factor (HCF) of 12600 and 14112 given their prime factorizations:
$12600=2^{3} \times 3^{2} \times 5^{2} \times 7$ and $14112=2^{5} \times 3^{2} \times 7^{2}$ .

Find the sum of all possible values for the number of factors $d$ of $m \times n$ , where $m$ and $n$ each have 6 factors.

Find the sum of all possible values of $d$ , where $d$ is the number of factors of $m \times n$ , given $m$ and $n$ each have 6 factors.

Find the factor pairs of 54 and identify the pair with a specific sum. Use $54 = a \times b$ .

Find all factors of 99. The factors are: $1, 3, 9, 11, 33, 99$ .

Identify the correct prime factorization of 504 in exponential form: $2^{3} \cdot 3^{2} \cdot 7^{2}$ , $2 \cdot 3 \cdot 7$ , $2 \cdot 3^{2} \cdot 7$ , or $2^{3} \cdot 3^{2} \cdot 7$ .

Find the GCF of 120 and 750 using prime factorization. The GCF is $\square$ .

Factor 30 into prime factors using a factor tree if needed. $30=\square \times \square \times \square$

Find the sum of all possible values for the number of factors $d$ of $m \times n$ , where $m$ and $n$ each have 6 factors.

Find the prime factorization of 44. Write it as a product of prime factors: $44 =$

Find the HCF of 70 and 385 using prime factorization. Correct the factors: 70 = $2 \times 5 \times 7$ , 385 = $5 \times 7 \times 11$ .

A spinner numbered 1 to 8 is spun. Find the probabilities for these events:
a. Factor of 35. b. Multiple of 4. c. Even. d. 3 or 8. e. 9. f. Composite. g. Neither prime nor composite.

Find $\log (1,000,000)$ without a calculator. $\log (1,000,000)=\square($ Type an integer or a decimal. $)$

Write 30 as a sum of two prime numbers. Choose from: A. $30=28+\square$ , B. $30=19+\square$ , C. $30=2+\square$ .

Determine if 7 is a Sophie Germain prime by calculating $2p + 1$ and checking if it's prime. What is $2p + 1 = \square$ ?

Find the number of divisors of $7^{8} \cdot 19^{2}$ . There are $\square$ divisors of this composite number.

Determine if the formula $n! \pm 1$ yields a prime for $n = 2, 3, 4, 5$ . Does it produce primes never, sometimes, or always?

Find the sixth Fermat number: $2^{2^{5}}+1$ . Then, divide it by 641. What are the results?

Is $35,390,008$ a happy number? Yes/No Is $39,259,592$ a happy number? Yes/No Are $35,390,008$ and $39,259,592$ a happy amicable pair? Yes/No

Find the next two primes of the form $4k+1$ after $5, 13, 17, 29$ and express them as sums of squares.

Find the GCF and LCM of $2^{32} \cdot 5^{17} \cdot 7^{22}$ and $2^{35} \cdot 5^{22} \cdot 7^{14}$ .

Find the prime numbers between 100 and $200$ . How many are there? Choices: A. 24 B. 23 C. 21 D. 19

Find the highest common factor (HCF) of $A = 2^2 \times 3^4 \times 7$ and $B = 3^2 \times 7^2$ .

Find the HCF and LCM of $A = 2^{2} \times 3^{4} \times 7$ and $B = 3^{2} \times 7^{2}$ .

Are the numbers $7,276,766,187$ and 4 relatively prime? Answer: Yes or No.

Show that $7,276,766,187$ and 4 are relatively prime by finding the GCD or checking divisibility.

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

Si $p$ est un nombre premier, est-ce que $p^{2}$ est également un nombre premier?

Find the probability of spinning a factor of 15 on an 8-sector spinner numbered 1 to 8. $P($ factor of 15 $)=\square$

Find a value for $a$ such that $a \bmod 5 = 3$ . How many values of $a$ satisfy this condition?

Find a value for $n$ such that $108 \bmod n = 0$ . How many such values for $n$ exist?

Find the LCM for each pair: (8, 12), (2, 6), (3, 8), (6, 8), (3, 4). Options: $6, 12, 24, 48$ .

Identify the correct prime factorization of 200 in exponential form: $2^{2} \cdot 5^{2}$ , $2 \cdot 10^{2}$ , $8 \cdot 25$ , or $2^{3} \cdot 5^{2}$ .