Number Theory

Problem 1101

List all positive factors of 12.

See Solution

Problem 1102

Find all positive factors of 49.

See Solution

Problem 1103

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

See Solution

Problem 1104

Find the LCM of $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ and $2^{2} \cdot 3^{3} \cdot 7 \cdot 13$ . Provide a single number.

See Solution

Problem 1105

Find all the positive factors of 40.

See Solution

Problem 1106

Find the square roots of 81. Options: A. 3 B. 9.5 C. -3 D. -9 E. $|9|$ F. 9

See Solution

Problem 1107

Mrs. Fisher has 91 watches. Can they be arranged in multiple rows and columns? Who is right, Mrs. or Mr. Fisher?

See Solution

Problem 1108

Mr. Deets can create arrays for 9 pictures. How many unique arrays can he form using different factor pairs? Odd or even?

See Solution

Problem 1109

Identify all irrational numbers from the set: $\{\pi_{0}-\sqrt{3},-0.5,0, \frac{1}{7}, \sqrt{5}, \sqrt{9}, 3 . \overline{3}\}$ . Options: $\pi,-\sqrt{3}, \sqrt{5}, \sqrt{9}$ ; $-\sqrt{3}, \sqrt{5}, \sqrt{9}, 3 . \overline{3}$ ; $\pi,-\sqrt{3}, \sqrt{5}$ ; $-\sqrt{3}, \sqrt{5}, 3 . \overline{3}$ .

See Solution

Problem 1110

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

See Solution

Problem 1111

Find the HCF and LCM of 60 and 220 using prime factorizations.

See Solution

Problem 1112

Find the HCF and LCM of 60 and 220 using their prime factors: 60 = $2^2 \cdot 3 \cdot 5$ , 220 = $2^2 \cdot 5 \cdot 11$ .

See Solution

Problem 1113

Find the HCF and LCM of 84 ( $2^2 \times 3 \times 7$ ) and 308 ( $2^2 \times 7 \times 11$ ).

See Solution

Problem 1114

Factor the number 54 into its prime factors.

See Solution

Problem 1115

Classify the numbers $5, 18, 27, 17$ as prime or composite.

See Solution

Problem 1116

List all factors of 36.

See Solution

Problem 1117

Find two prime numbers that sum to 28. What is their difference?

See Solution

Problem 1118

How can 256 be expressed using 4? Choose all that fit. A. $4+4+4+4$ B. $4^{4}$ C. $4 \times 4 \times 4 \times 4 \times 4$ D. $4 \times 4 \times 4 \times 4$

See Solution

Problem 1119

Express 75 as a product of its prime factors. $75 = \square$

See Solution

Problem 1120

Factor 50 into its prime components. $50=2 \times 5^2$

See Solution

Problem 1121

Identify the prime numbers from this list: 14, 19, 3, 8.

See Solution

Problem 1122

Identify the prime numbers from the list: 1, 2, 7, 15.

See Solution

Problem 1123

Identify the prime number from the list: 5, 4, 16, 15.

See Solution

Problem 1124

Identify the prime number from the list: 3, 36, 55, 44.

See Solution

Problem 1125

Identify all prime numbers from the list: 17, 5, 11, 7.

See Solution

Problem 1126

Identify the prime number from this list: 14, 12, 17, 6.

See Solution

Problem 1127

Select a number from 1 to 18. Find the probability for: (a) odd, (b) even, (c) prime, (d) prime and odd, (e) prime and even.

See Solution

Problem 1128

What is the ratio of the value of the 5 in 9,503 to the value of the 5 in 17.750? Calculate $\frac{5 \times 100}{5 \times 10}$ .

See Solution

Problem 1129

Find the prime decomposition of 1197 in index form using its prime factor tree.

See Solution

Problem 1130

Identify the prime numbers from this list and mark them: 2, 6, 10, 13, 18, 25, or None of the above.

See Solution

Problem 1131

Check which numbers are prime: 5, 10, 22, 25, 27, 28, or "None of the above."

See Solution

Problem 1132

Check the prime numbers: 6, 7, 15, 22, 24, 29, or "None of the above".

See Solution

Problem 1133

Check which numbers are prime: 7, 9, 10, 11, 17, 29, or None of the above.

See Solution

Problem 1134

Prove that the sum of three consecutive integers is divisible by 3 using deductive reasoning.

See Solution

Problem 1135

Avery claims the GCF of 12 and 15 is 60. Is this true? Justify your answer.

See Solution

Problem 1136

Find the prime factorization of 18. Is it prime?

See Solution

Problem 1137

Find the tenth triangular number in the sequence $1, 3, 6, 10, \ldots$ which is formed by $1 + 2 + 3 + \ldots + n$ .

See Solution

Problem 1138

Find the prime factorization of 110. Options: A) $5^{2} \cdot 11$ B) $2^{5} \cdot 11$ C) $5 \cdot 22$ D) $2 \cdot 5 \cdot 11$

See Solution

Problem 1139

Find the Lowest Common Multiple (LCM) of $A = 3 \times 5^{2}$ and $B = 2^{2} \times 3^{2} \times 7$ .

See Solution

Problem 1140

What defines an irrational number?
1. Decimal that repeats and does not terminate
2. Decimal that neither repeats nor terminates
3. Square root not resulting in a whole number
4. Decimal that terminates and does not repeat

See Solution

Problem 1141

Kennedy's wall is $16 \mathrm{in}.$ high and $28 \mathrm{in}.$ long. What is the largest square tile side length? How many tiles needed?

See Solution

Problem 1142

Kennedy's wall is $16 \mathrm{in}$ high and $28 \mathrm{in}$ long. Find the largest square tile side length that fits.

See Solution

Problem 1143

Kennedy's wall is 16 in high and 28 in long.
a. Find the largest square tile side length. b. Calculate how many tiles are needed to cover the wall.

See Solution

Problem 1144

Is Avery correct that the GCF of 12 and 15 is 60? What is the GCF of 60 and 100? Show your work.

See Solution

Problem 1145

Avery claims the GCF of 12 and 15 is 60. Is this true? Justify your answer.

See Solution

Problem 1146

Find twin primes between 32 and 41, and identify any pairs that exist.

See Solution

Problem 1147

Identify all natural number factors of 110.

See Solution

Problem 1148

Express the number 44 as a sum of two prime numbers.

See Solution

Problem 1149

Is it true or false that if $\mathrm{n}$ is composite, then $2^{n}-1$ is also composite?

See Solution

Problem 1150

Is it true or false that if a natural number is divisible by 7 and 2, it is also divisible by 14?

See Solution

Problem 1151

Is it true or false that if $\mathrm{n}$ is composite, then $2^{n}-1$ is also composite?

See Solution

Problem 1152

Determine if the statement is true or false: If a number is divisible by both 3 and 9, is it also divisible by 27?

See Solution

Problem 1153

Find the prime factorization of 2013 using exponents where applicable.

See Solution

Problem 1154

Find the prime factorization of 126 using exponents where applicable.

See Solution

Problem 1155

Find the probability of selecting a prime number from the set $\{1,2,3,\ldots,15\}$ . Answer: $\square$ .

See Solution

Problem 1156

Mrs. Fisher has 91 watches. Can they be arranged in rows and columns evenly? Who's right: Mrs. or Mr. Fisher?

See Solution

Problem 1157

Find an odd number that is composite and explain why it is composite.

See Solution

Problem 1158

Which number is not a factor of both 36 and 84? (A) 2 (B) 3 (C) 5 (D) 6. How many coins gives 2 arrays? (A) 10 (B) 16 (C) 25 (D) 29.

See Solution

Problem 1159

Find the prime factorization of $924$ . Choose from: a. $2 \times 2 \times 21 \times 11$ , b. $2 \times 2 \times 3 \times 7 \times 11$ , c. $2 \times 3 \times 3 \times 7 \times 11$ , d. $4 \times 3 \times 7 \times 11$ .

See Solution

Problem 1160

Jason can fill $\square$ bags using the GCF of 26 and 39. Each bag has $\square$ strawberry and $\square$ raspberry scones.

See Solution

Problem 1161

Find the GCF of the prime numbers 43 and 47. The GCF is $\square$ . (Type a whole number.)

See Solution

Problem 1162

Find the rule for divisibility by 9 and the digit $a$ in $27a8$ for it to be divisible by 9.

See Solution

Problem 1163

Montrez que tout entier est congru à la somme de ses chiffres modulo 3 après avoir étudié les puissances de 10 modulo 3. Quelle règle en découle ?

See Solution

Problem 1164

Calculer et raisonner sur la divisibilité par 3 : montrer que tout entier est congru à la somme de ses chiffres modulo 3. Quelle règle en découle ?

See Solution

Problem 1165

Find the greatest common factor of $5a^{3}$ and 11.

See Solution

Problem 1166

Determine if each statement is true or false. Provide a division equation for proof.
1. $5 \mid 15$
2. $4 \mid 36$
3. $6 \mid 39$
4. $8 \mid 40$
5. $9 \mid 82$
6. $6 \mid 46$
7. $3 \mid 90$
8. $5 \mid 85$
9. $4 \mid 128$
10. $8 \mid 374$
11. $2 \mid 577$
12. $4 \mid 634$
13. $9 \mid 567$
14. $7 \mid 473$
15. $8 \mid 688$

See Solution

Problem 1167

List all prime numbers in the range from 0 to 10.

See Solution

Problem 1168

Identify the prime numbers from this list: 29, 31, 47, 51. Select all that apply.

See Solution

Problem 1169

Determine if the following numbers are factors and their divisibility:
1. Is 3 a factor of 47? Calculate $47 \div 3$ .
2. Is 96 divisible by 13? Calculate $96 \div 13$ .
3. Is 5 a factor of 15? Calculate $15 \div 5$ .

See Solution

Problem 1170

Test if 706, 9459, and 25915 are divisible by 3. Find the next number to check for each.

See Solution

Problem 1171

Determine if the following numbers are factors or divisible. Use $a \div b = c R d$ format for answers.
1. Is 3 a factor of 50?
2. Is 95 divisible by 15?
3. Is 5 a factor of 15?

See Solution

Problem 1172

Find the prime factorizations of the numbers 74 and 55 in expanded form and exponential form. Use $*$ for multiplication and "^" for exponents.

See Solution

Problem 1173

Complete the table: Is 14 divisible by $2$ ? $14 \div 2=\square R$ . Is 12 a factor of $103$ ? $103 \div 12=$ ? Is 7 a factor of 14? $14 \div 7=\square R$ .

See Solution

Problem 1174

Find the prime factorizations of the numbers in both expanded form and exponential form. Use $*$ for multiplication and "^" for exponents.

See Solution

Problem 1175

Find the prime factorizations of the numbers in expanded and exponential forms. Example: 44 → $2 \cdot 2 \cdot 11$ and $2^{2} \cdot 11$ .

See Solution

Problem 1176

List the first 10 multiples of 8 and 12, then find their common multiples and the smallest one.

See Solution

Problem 1177

Find the prime factorizations of the numbers in expanded and exponential forms, e.g., for 44: $2 \cdot 2 \cdot 11$ and $2^{2} \cdot 11$ .

See Solution

Problem 1178

Find the least common factor (LCF) for these pairs: a. 36 and 24 b. 49 and 70 c. 1 and 128 d. 32, 26, and 28 e. $m$ and $n$ . f. Is finding LCF useful? (yes/no) g. What is the LCF of any set of numbers?

See Solution

Problem 1179

Find the prime factorizations of the numbers in expanded form and exponential form. Example: 44 is $2 \cdot 2 \cdot 11$ and $2^{2} \cdot 11$ .

See Solution

Problem 1180

Find the prime factorizations of the numbers. Show both expanded form ( $a \cdot b \cdots$ ) and exponential form ( $a^{\wedge} b$ ).

See Solution

Problem 1181

Choose the prime factorization of 100: (A) $4 \times 25$ , (B) $2^{2} \times 5^{2}$ , (C) $2^{2} \times 2^{5}$ , (D) $10 \times 10$ .

See Solution

Problem 1182

Select the correct statements about the pairs: A: 56,98; B: 54,66; C: 63,72. GCFs and LCMs involved.

See Solution

Problem 1183

Select all prime numbers from this list: 99, 102, 7, 87.

See Solution

Problem 1184

Find the remainder of $17^{234}$ when divided by 31.

See Solution

Problem 1185

What is the remainder of $17^{234}$ when divided by 31?

See Solution

Problem 1186

Find the remainder of $17^{234}$ when divided by 31.

See Solution

Problem 1187

Beweise, dass $\sqrt[4]{8}$ irrational ist und erkläre, warum $\sqrt[3]{216}$ nicht so bewiesen werden kann.

See Solution

Problem 1188

Is 52 a perfect square? Justify your answer using its factors or properties of perfect squares.

See Solution

Problem 1189

Find the remainder when $181$ is divided by $7$ . Explain your method. Then, verify your answer using the bus stop method.

See Solution

Problem 1190

If 33 divides $n$ , what are the other natural numbers that divide $n$ ? Explain your reasoning.
The other natural numbers that divide $n$ are $\square$ .

See Solution

Problem 1191

Determine if the number $5$ is prime or composite.

See Solution

Problem 1192

Find the prime factorization of 36 and 45. Also, determine the GCF and LCM of 36 and 45.

See Solution

Problem 1193

A prime number has how many factors? Justify your answer using the number 80.

See Solution

Problem 1194

Find all composite integers $n>1$ such that each divisor $d_i$ divides $d_{i+1}+d_{i+2}$ for all $1 \leq i \leq k-2$ .

See Solution

Problem 1195

Is 159 an even number, composite number, prime number, or a proper fraction?

See Solution

Problem 1196

Complete the prime factorization for 222 and 648:
$222=2 \times \text{ }$
$648=$
3.

See Solution

Problem 1197

What is the prime number outcome when rolling a 6-sided die?

See Solution

Problem 1198

What is the probability $\mathrm{P}$ of picking a prime or divisor of 14 from a deck? Simplify your answer.

See Solution

Problem 1199

What is the probability $P(\text{not divisor of } 72)$ when picking a card at random? Simplify your answer.

See Solution

Problem 1200

Determine the prime factorization of 945. Choose from: (A) $3 \cdot 5 \cdot 7$ , (B) $3^{2} \cdot 5 \cdot 7$ , (C) $3^{3} \cdot 5 \cdot 7$ , (D) $3^{3} \cdot 5$ .

See Solution

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Number Theory

Problem 1101

List all positive factors of 12.

Problem 1102

Find all positive factors of 49.

Problem 1103

Problem 1104

Find the LCM of 24⋅32⋅5⋅722^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}24⋅32⋅5⋅72 and 22⋅33⋅7⋅132^{2} \cdot 3^{3} \cdot 7 \cdot 1322⋅33⋅7⋅13. Provide a single number.

Problem 1105

Find all the positive factors of 40.

Problem 1106

Find the square roots of 81. Options: A. 3 B. 9.5 C. -3 D. -9 E. ∣9∣|9|∣9∣ F. 9

Problem 1107

Mrs. Fisher has 91 watches. Can they be arranged in multiple rows and columns? Who is right, Mrs. or Mr. Fisher?

Problem 1108

Mr. Deets can create arrays for 9 pictures. How many unique arrays can he form using different factor pairs? Odd or even?

Problem 1109

Problem 1110

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

Problem 1111

Find the HCF and LCM of 60 and 220 using prime factorizations.

Problem 1112

Find the HCF and LCM of 60 and 220 using their prime factors: 60 = 22⋅3⋅52^2 \cdot 3 \cdot 522⋅3⋅5, 220 = 22⋅5⋅112^2 \cdot 5 \cdot 1122⋅5⋅11.

Problem 1113

Find the HCF and LCM of 84 (22×3×72^2 \times 3 \times 722×3×7) and 308 (22×7×112^2 \times 7 \times 1122×7×11).

Problem 1114

Factor the number 54 into its prime factors.

Problem 1115

Classify the numbers 5,18,27,175, 18, 27, 175,18,27,17 as prime or composite.

Problem 1116

List all factors of 36.

Problem 1117

Find two prime numbers that sum to 28. What is their difference?

Problem 1118

How can 256 be expressed using 4? Choose all that fit. A. 4+4+4+44+4+4+44+4+4+4 B. 444^{4}44 C. 4×4×4×4×44 \times 4 \times 4 \times 4 \times 44×4×4×4×4 D. 4×4×4×44 \times 4 \times 4 \times 44×4×4×4

Problem 1119

Express 75 as a product of its prime factors. 75=□ 75 = \square 75=□

Problem 1120

Factor 50 into its prime components. 50=2×52 50=2 \times 5^2 50=2×52

Problem 1121

Identify the prime numbers from this list: 14, 19, 3, 8.

Problem 1122

Identify the prime numbers from the list: 1, 2, 7, 15.

Problem 1123

Identify the prime number from the list: 5, 4, 16, 15.

Problem 1124

Identify the prime number from the list: 3, 36, 55, 44.

Problem 1125

Identify all prime numbers from the list: 17, 5, 11, 7.

Problem 1126

Identify the prime number from this list: 14, 12, 17, 6.

Problem 1127

Select a number from 1 to 18. Find the probability for: (a) odd, (b) even, (c) prime, (d) prime and odd, (e) prime and even.

Problem 1128

What is the ratio of the value of the 5 in 9,503 to the value of the 5 in 17.750? Calculate 5×1005×10\frac{5 \times 100}{5 \times 10}5×105×100​.

Problem 1129

Find the prime decomposition of 1197 in index form using its prime factor tree.

Problem 1130

Identify the prime numbers from this list and mark them: 2, 6, 10, 13, 18, 25, or None of the above.

Problem 1131

Check which numbers are prime: 5, 10, 22, 25, 27, 28, or "None of the above."

Problem 1132

Check the prime numbers: 6, 7, 15, 22, 24, 29, or "None of the above".

Problem 1133

Check which numbers are prime: 7, 9, 10, 11, 17, 29, or None of the above.

Problem 1134

Prove that the sum of three consecutive integers is divisible by 3 using deductive reasoning.

Problem 1135

Avery claims the GCF of 12 and 15 is 60. Is this true? Justify your answer.

Problem 1136

Find the prime factorization of 18. Is it prime?

Problem 1137

Find the tenth triangular number in the sequence 1,3,6,10,…1, 3, 6, 10, \ldots1,3,6,10,… which is formed by 1+2+3+…+n1 + 2 + 3 + \ldots + n1+2+3+…+n.

Problem 1138

Find the prime factorization of 110. Options: A) 52⋅115^{2} \cdot 1152⋅11 B) 25⋅112^{5} \cdot 1125⋅11 C) 5⋅225 \cdot 225⋅22 D) 2⋅5⋅112 \cdot 5 \cdot 112⋅5⋅11

Problem 1139

Find the Lowest Common Multiple (LCM) of A=3×52A = 3 \times 5^{2}A=3×52 and B=22×32×7B = 2^{2} \times 3^{2} \times 7B=22×32×7.

Problem 1140

What defines an irrational number? 1. Decimal that repeats and does not terminate 2. Decimal that neither repeats nor terminates 3. Square root not resulting in a whole number 4. Decimal that terminates and does not repeat

Problem 1141

Find the LCM of $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ and $2^{2} \cdot 3^{3} \cdot 7 \cdot 13$ . Provide a single number.

Find the square roots of 81. Options: A. 3 B. 9.5 C. -3 D. -9 E. $|9|$ F. 9

Find the HCF and LCM of 60 and 220 using their prime factors: 60 = $2^2 \cdot 3 \cdot 5$ , 220 = $2^2 \cdot 5 \cdot 11$ .

Find the HCF and LCM of 84 ( $2^2 \times 3 \times 7$ ) and 308 ( $2^2 \times 7 \times 11$ ).

Classify the numbers $5, 18, 27, 17$ as prime or composite.

How can 256 be expressed using 4? Choose all that fit. A. $4+4+4+4$ B. $4^{4}$ C. $4 \times 4 \times 4 \times 4 \times 4$ D. $4 \times 4 \times 4 \times 4$

Express 75 as a product of its prime factors. $75 = \square$

Factor 50 into its prime components. $50=2 \times 5^2$

What is the ratio of the value of the 5 in 9,503 to the value of the 5 in 17.750? Calculate $\frac{5 \times 100}{5 \times 10}$ .

Find the tenth triangular number in the sequence $1, 3, 6, 10, \ldots$ which is formed by $1 + 2 + 3 + \ldots + n$ .

Find the prime factorization of 110. Options: A) $5^{2} \cdot 11$ B) $2^{5} \cdot 11$ C) $5 \cdot 22$ D) $2 \cdot 5 \cdot 11$

Find the Lowest Common Multiple (LCM) of $A = 3 \times 5^{2}$ and $B = 2^{2} \times 3^{2} \times 7$ .

What defines an irrational number?
1. Decimal that repeats and does not terminate
2. Decimal that neither repeats nor terminates
3. Square root not resulting in a whole number
4. Decimal that terminates and does not repeat

Kennedy's wall is $16 \mathrm{in}.$ high and $28 \mathrm{in}.$ long. What is the largest square tile side length? How many tiles needed?

Kennedy's wall is $16 \mathrm{in}$ high and $28 \mathrm{in}$ long. Find the largest square tile side length that fits.

Kennedy's wall is 16 in high and 28 in long.
a. Find the largest square tile side length. b. Calculate how many tiles are needed to cover the wall.

Is it true or false that if $\mathrm{n}$ is composite, then $2^{n}-1$ is also composite?

Is it true or false that if $\mathrm{n}$ is composite, then $2^{n}-1$ is also composite?

Find the probability of selecting a prime number from the set $\{1,2,3,\ldots,15\}$ . Answer: $\square$ .

Jason can fill $\square$ bags using the GCF of 26 and 39. Each bag has $\square$ strawberry and $\square$ raspberry scones.

Find the GCF of the prime numbers 43 and 47. The GCF is $\square$ . (Type a whole number.)

Find the rule for divisibility by 9 and the digit $a$ in $27a8$ for it to be divisible by 9.

Find the greatest common factor of $5a^{3}$ and 11.

Determine if the following numbers are factors and their divisibility:
1. Is 3 a factor of 47? Calculate $47 \div 3$ .
2. Is 96 divisible by 13? Calculate $96 \div 13$ .
3. Is 5 a factor of 15? Calculate $15 \div 5$ .

Determine if the following numbers are factors or divisible. Use $a \div b = c R d$ format for answers.
1. Is 3 a factor of 50?
2. Is 95 divisible by 15?
3. Is 5 a factor of 15?

Find the prime factorizations of the numbers 74 and 55 in expanded form and exponential form. Use $*$ for multiplication and "^" for exponents.

Complete the table: Is 14 divisible by $2$ ? $14 \div 2=\square R$ . Is 12 a factor of $103$ ? $103 \div 12=$ ? Is 7 a factor of 14? $14 \div 7=\square R$ .

Find the prime factorizations of the numbers in both expanded form and exponential form. Use $*$ for multiplication and "^" for exponents.