Number Theory

Problem 201

Find the GCF of 28 and 35, then calculate $\frac{35}{1.35}$ . Finally, find the GCF of 40 and 240.

See Solution

Problem 202

Classify a real number as natural, integer, rational, or irrational. Can a square root be rational?

See Solution

Problem 203

Determine the truth value of the converse: If $n$ is an odd natural number > 6, then $n$ is prime. Counterexample if false.

See Solution

Problem 204

Identify the true statement about rational and irrational numbers.

See Solution

Problem 205

Identify the number where 4 represents 400 from the set: 142568, 514682, 821456, 64251.

See Solution

Problem 206

Find the missing number in each prime factorization: a) $200=2 \times 2 \times 1 \times 5 \times 5$ b) $216=2^{3} \times 3$ c) $8281=7 \times 7 \times 13 \times$ d) $1568=5 \times 7^{2}$

See Solution

Problem 207

You have 150 chairs for a play. How many ways can you arrange them in equal rows? There are 12 arrangements. Are they all suitable? Explain.

See Solution

Problem 208

Find the sum of the first five prime numbers: 2, 3, 5, 7, and 11. What is the result?

See Solution

Problem 209

Organize 27 grade 6, 36 grade 7, and 54 grade 8 students into equal groups. Find the largest group size and number of groups.

See Solution

Problem 210

Find the smallest value of $\frac{p}{q}$ where $p$ is a common multiple of 4 and 9, and $q$ is a common factor of 72 and 120.

See Solution

Problem 211

Express 120 as a product of prime factors in index form.

See Solution

Problem 212

What is the maximum length in metres of equal pieces Natalie can cut from 26 m of blue wire and 12 m of green wire?

See Solution

Problem 213

Express 192 as a product of its prime factors.

See Solution

Problem 214

Express 600 as $2^{a} \times b \times c^{d}$ with $a, b, c, d$ as prime numbers. Determine $a, b, c, d$ .

See Solution

Problem 215

Identify the prime number from the list: 48, 21, 19.

See Solution

Problem 216

Identify which of Dominic's factor pairs for 120 are both composite: A. $40 \times 3$ , B. $60 \times 2$ , C. $10 \times 12$ , D. $5 \times 24$ .

See Solution

Problem 217

Lila claims 123 is composite. Which statement proves her right? A. All odd numbers are composite. B. 123 has a factor of 3. C. 123 is divisible by 4. D. 123 is a factor of 246.

See Solution

Problem 218

Explain the four-step process for solving problems with large numbers (billions), including place-value and representations.

See Solution

Problem 219

Find the integer-sided rectangles with an area of 36 square units, including squares, allowing similar rectangles.

See Solution

Problem 220

Classify each number as prime or composite: 14, 11, 9, 21, 23. Place a check in the correct box for each.

See Solution

Problem 221

Is 1 a prime or composite number? Provide an explanation for your answer.

See Solution

Problem 222

Which number is prime: 1, 48, 21, or 19?

See Solution

Problem 223

Is the number 1 prime or composite? Explain your reasoning.

See Solution

Problem 224

What conclusion about the number 30 can be drawn from its factor pairs: 1, 2, 3, 5? A. Prime or composite? B. Prime? C. No prime factors? D. Composite?

See Solution

Problem 225

Identify a prime factor of the composite number 58: A. 1, B. 7, C. 8, D. 29.

See Solution

Problem 226

Express each number as powers in two different ways: a) 16 b) 81 c) 256 d) 1024 Examples: $4^{2}$ , $2^{4}$ for 16.

See Solution

Problem 227

Find the prime factorization of 500. Is it $2^{2} 5^{3}$ , $2^{2} 3^{5}$ , $2^{2} 50$ , or $5 * 100$ ?

See Solution

Problem 228

Find the prime factorization of 12. Options: $2^{6}$ , $2^{2} \cdot 3$ , $2^{4}$ , $8 \cdot 8$ , $2 \cdot 6$ .

See Solution

Problem 229

Find the prime factorization of 18. Options: $5^{9}$ , $2 \cdot 3^{2}$ , $5 \cdot 9$ , $15^{3}$ .

See Solution

Problem 230

Identify the factors of 21 from the list: 4, 12, 7, 3, 5, 2.

See Solution

Problem 231

Why is there a coefficient of 3 in $\sqrt[3]{54} = 3 \sqrt[3]{2}$ ? Consider the factors of 54.

See Solution

Problem 232

Aus einer Urne mit 100 Kugeln: Bestimmen Sie die Ergebnisse und Wahrscheinlichkeiten für die Ereignisse A, B, C, D, E, F.

See Solution

Problem 233

Identify which numbers are factors of 19,656, given its prime factorization $2^{3} \times 3^{3} \times 7 \times 13$ . Options: 297, 16, 14.

See Solution

Problem 234

Find the factors of 16,632 from these numbers based on their prime factorizations:
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 235

Find the prime factorization of 1683 in index form using the given prime factor tree.

See Solution

Problem 236

Which of the following numbers are factors of 19,656 ( $2^{3} \times 3^{3} \times 7 \times 13$ )? Select all that apply: $8=2^{3}$ , $297=3^{3} \times 11$ , $63=3^{2} \times 7$ , $16=2^{4}$ , $14=2 \times 7$ .

See Solution

Problem 237

Factor 132 into its prime components.

See Solution

Problem 238

Find the possible row arrangements for 16 cheerleaders, with each row having the same number of members.

See Solution

Problem 239

Find the smallest positive integer to multiply 360 by to make it a perfect square. Use prime factorization to explain.

See Solution

Problem 240

If 77 divides $n$ , what are the other natural numbers that also divide $n$ ? List them.

See Solution

Problem 241

Casper's number has an HCF of 4 with 16. a) Find the smallest possible number. b) List two other possible values.

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Problem 242

Liste die Elemente der Mengen: a) $A=\{x \mid x$ ist eine Primzahl und $x > 3\}$ , b) $B=\{x \mid x$ ist eine Primzahl und $x < 20\}$ .

See Solution

Problem 243

Find the prime factorization of 18 using exponents.

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Problem 244

Complete one column of fluency facts each night without a calculator. Circle all PRIME numbers: 11, 15, 23, 88, 77, 62, 103, 4.

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Problem 245

Find the GCF of 42 and 60 using prime factorization. Show all your work.

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Problem 246

Find the GCF of 30 and 40 using prime factorization. Show all your work.

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Problem 247

Find two factors of 1001 using the factors 7 and 13.

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Problem 248

Find the largest perfect square that divides 338.

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Problem 249

Which option shows the prime factorization of 200 in exponential form? $2^{2} \cdot 5^{2}$ , $8 \cdot 25$ , $2^{3} \cdot 5^{2}$ , or $2 \cdot 10^{2}$ ?

See Solution

Problem 250

Find a prime number between 30 and 80 whose digits sum to 14.

See Solution

Problem 251

Find a prime number between 30 and 80 whose digits sum to 14.

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Problem 252

Respond to a student claiming $188_{\text{ten}}$ is $723_{\text{five}}$ by showing $7 \times 5^{2}+2 \times 5^{1}+3 \times 5^{0}=18$ .

See Solution

Problem 253

Beurteile, ob die Aussagen wahr oder falsch sind: a) "Es gibt unendlich viele Punkte auf der Zahlengeraden, die rationalen Zahlen entsprechen." b) "Es gibt unendlich viele Punkte auf der Zahlengeraden, die irrationalen Zahlen entsprechen."

See Solution

Problem 254

1. Wandle 218 in Binär- und Hexadezimalzahl um.
2. Was ist die Zahl (B1.4) im Dezimalsystem?
3. Subtrahiere $b=01100101$ von $a=01001110$ im Zweierkomplement und gib das Ergebnis als Dezimalzahl an.

See Solution

Problem 255

Find common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 1120 ( $2^{5} \times 5 \times 7$ ) from the options given.

See Solution

Problem 256

Find the common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 1120 ( $2^{5} \times 5 \times 7$ ). Select all correct answers.

See Solution

Problem 257

Create prime factor trees for 90 and 135, then use them to calculate the LCM of 90 and 135.

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Problem 258

Create prime factor trees for 98 and 165, then find their lowest common multiple (LCM).

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Problem 259

Find the HCF and LCM of 60 ( $2^2 \times 3 \times 5$ ) and 220 ( $2^2 \times 5 \times 11$ ).

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Problem 260

1a. Factor 60 and 36 into primes. 1b. Find the LCM of 60 and 36. 1c. If moons Anderon (60 days) and Barberon (36 days) align on 1 March 2023, when will they next align?
2. Each letter is coded as $A=1, B=2, \ldots, Z=26$ . For a 3-letter word, find the word that equals 7500 when coded.

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Problem 261

Create prime factor trees for 30 and 170, then determine their highest common factor.

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Problem 262

Draw the prime factor trees for 42 and 66. Use them to find the LCM of 42 and 66.

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Problem 263

Find the prime factorization of 60 and express it in index form: $60 = 2^a \times 3^b \times 5^c$ .

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Problem 264

Create prime factor trees for 30 and 110 to determine their highest common factor (HCF).

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Problem 265

a) Nenne drei Gruppen von drei aufeinanderfolgenden natürlichen Zahlen, die Vielfache einer Quadratzahl > 1 sind. b) Beweise, dass es unendlich viele Gruppen von drei aufeinanderfolgenden natürlichen Zahlen gibt, die Vielfache einer Quadratzahl > 1 sind. c) Finde ein Beispiel mit vier aufeinanderfolgenden natürlichen Zahlen, die Vielfache einer Quadratzahl > 1 sind.

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Problem 266

Finlay has 15 calculators and 18 rulers. What's the max number of equal boxes he can create for both items?

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Problem 267

Find the LCM of 12 and 15. What inputs produce outputs of 12 and 42?

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Problem 268

Find the prime number from the list: 75, 56, 43, 63, 34. Use divisibility tests to help identify it.

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Problem 269

Determine if the following numbers are prime: 81, 57, 53.

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Problem 270

Beurteile, ob die Aussagen wahr oder falsch sind: a) Unendlich viele rationale Zahlen auf der Zahlengeraden? b) Unendlich viele irrationale Zahlen auf der Zahlengeraden?

See Solution

Problem 271

Find the highest common factor (HCF) of $A=2^{3} \times 3 \times 5$ and $B=2^{2} \times 3 \times 5^{2}$ .

See Solution

Problem 272

Bestimme die Teilermengen für die folgenden Zahlen: $T_{36}$ , $T_{63}$ , $T_{56}$ , $T_{81}$ , $T_{100}$ , $T_{85}$ , $T_{37}$ , $T_{243}$ , $T_{225}$ .

See Solution

Problem 273

Find the HCF of $G = 2^5 \times 3^3 \times 5$ and $H = 2^2 \times 3^6 \times 5 \times 7$ . Express in prime factor form.

See Solution

Problem 274

Find the prime factor decomposition of 208 using $2^{3} \times 13$ for 104. Provide your answer in index form.

See Solution

Problem 275

Find the prime decomposition of 1197 in index form using the factor tree: $1197 = 3^2 \times 7^1 \times 19^1$ .

See Solution

Problem 276

Find the prime factorization of 300 and express it in index form.

See Solution

Problem 277

Write 600 as a product of prime factors in index form.

See Solution

Problem 278

Find the prime factorization of these numbers: a) 36, b) 80. Use $8\left(^{\star}\right)$ for multiplication.

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Problem 279

Find the prime factorization of $150$ . A. $3^{5} \times 5^{2}$ B. $2 \times 3^{2} \times 5$ C. $2 \times 3 \times 5^{2}$ D. $6 \times 5^{2}$

See Solution

Problem 280

Find the prime factorization of 57. Options: $5 \cdot 11.4$ , $3 \cdot 19$ , $3 \cdot 29$ , $2 \cdot 28.5$ .

See Solution

Problem 281

Find the prime factorization of 28. Options: $4 \cdot 7$ , $2^{2} \cdot 7$ , $2^{3} \cdot 7$ , $2^{2} \cdot 7^{2}$ .

See Solution

Problem 282

Find the prime factorization of 120. $120=$ $2^{3} \cdot 3 \cdot 5$

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Problem 283

Find the prime factorization of 43.

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Problem 284

Find prime factors of 27 and 90. List factors, then calculate LCM and GCF. Prime factors: $27 =$

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Problem 285

Find the prime factors of 27 and 90. Then, determine their LCM and GCF. List factors from least to greatest.

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Problem 286

Find the greatest common factor of $A=2 \cdot 5 \cdot 11 \cdot 23 \cdot 53$ and $B=3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 53$ .

See Solution

Problem 287

Find the LCM of $A = 3^2 \cdot 5 \cdot 7 \cdot 13$ and $B = 7^3 \cdot 13$ .

See Solution

Problem 288

Find the LCM of $A = 2 \cdot 3 \cdot 7^3$ and $B = 2 \cdot 7^2 \cdot 11$ .

See Solution

Problem 289

Find the LCM of the numbers $A = 3 \cdot 5 \cdot 7$ and $B = 3^2 \cdot 5 \cdot 7^2 \cdot 11$ .

See Solution

Problem 290

Find the GCF of $A = 3 \cdot 3 \cdot 5 \cdot 7 \cdot 13$ and $B = 7 \cdot 7 \cdot 7 \cdot 13$ .

See Solution

Problem 291

Find the GCF of $A = 2 \cdot 3 \cdot 7^3$ and $B = 2 \cdot 7^2 \cdot 11$ .

See Solution

Problem 292

Find the GCF of $A=2^4 \cdot 7$ and $B=2 \cdot 5^2 \cdot 11$ .

See Solution

Problem 293

Find the GCF of $A = 3 \cdot 3 \cdot 5 \cdot 7 \cdot 13$ and $B = 7 \cdot 7 \cdot 7 \cdot 13$ .

See Solution

Problem 294

List all factors of 36 and explain your process.

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Problem 295

Untersuche die Zahlen auf Teilbarkeit durch 2, 5 und 10: a) 90, 110, 225, 439, 653, 765, 825 b) 1258, 2270, 3280, 5301, 6475, 8500 c) 11075, 13406, 37895, 120003

See Solution

Problem 296

Rahul claims the $3$ s in 45,339 differ from the $6$ s in 66,084. How would you explain this?

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Problem 297

Find the LCM of 70 and 273 using their prime factor trees.

See Solution

Problem 298

Identify the three prime numbers from this list: 6, 2, 7, 19, 1, 15.

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Problem 299

Factor 715 into its prime components.

See Solution

Problem 300

Identify the correct prime factorization of 36 in exponential form from these options: $2 \cdot 3^{2}$ , $6^{2}$ , $4 \cdot 9$ , $2^{2} \cdot 3^{2}$ .

See Solution

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

Number Theory

Problem 201

Find the GCF of 28 and 35, then calculate 351.35\frac{35}{1.35}1.3535​. Finally, find the GCF of 40 and 240.

Problem 202

Classify a real number as natural, integer, rational, or irrational. Can a square root be rational?

Problem 203

Determine the truth value of the converse: If nnn is an odd natural number > 6, then nnn is prime. Counterexample if false.

Problem 204

Identify the true statement about rational and irrational numbers.

Problem 205

Identify the number where 4 represents 400 from the set: 142568, 514682, 821456, 64251.

Problem 206

Find the missing number in each prime factorization: a) 200=2×2×1×5×5200=2 \times 2 \times 1 \times 5 \times 5200=2×2×1×5×5 b) 216=23×3216=2^{3} \times 3216=23×3 c) 8281=7×7×13×8281=7 \times 7 \times 13 \times8281=7×7×13× d) 1568=5×721568=5 \times 7^{2}1568=5×72

Problem 207

You have 150 chairs for a play. How many ways can you arrange them in equal rows? There are 12 arrangements. Are they all suitable? Explain.

Problem 208

Find the sum of the first five prime numbers: 2, 3, 5, 7, and 11. What is the result?

Problem 209

Organize 27 grade 6, 36 grade 7, and 54 grade 8 students into equal groups. Find the largest group size and number of groups.

Problem 210

Find the smallest value of pq\frac{p}{q}qp​ where ppp is a common multiple of 4 and 9, and qqq is a common factor of 72 and 120.

Problem 211

Express 120 as a product of prime factors in index form.

Problem 212

What is the maximum length in metres of equal pieces Natalie can cut from 26 m of blue wire and 12 m of green wire?

Problem 213

Express 192 as a product of its prime factors.

Problem 214

Express 600 as 2a×b×cd2^{a} \times b \times c^{d}2a×b×cd with a,b,c,da, b, c, da,b,c,d as prime numbers. Determine a,b,c,da, b, c, da,b,c,d.

Problem 215

Identify the prime number from the list: 48, 21, 19.

Problem 216

Identify which of Dominic's factor pairs for 120 are both composite: A. 40×340 \times 340×3, B. 60×260 \times 260×2, C. 10×1210 \times 1210×12, D. 5×245 \times 245×24.

Problem 217

Lila claims 123 is composite. Which statement proves her right? A. All odd numbers are composite. B. 123 has a factor of 3. C. 123 is divisible by 4. D. 123 is a factor of 246.

Problem 218

Explain the four-step process for solving problems with large numbers (billions), including place-value and representations.

Problem 219

Find the integer-sided rectangles with an area of 36 square units, including squares, allowing similar rectangles.

Problem 220

Classify each number as prime or composite: 14, 11, 9, 21, 23. Place a check in the correct box for each.

Problem 221

Is 1 a prime or composite number? Provide an explanation for your answer.

Problem 222

Which number is prime: 1, 48, 21, or 19?

Problem 223

Is the number 1 prime or composite? Explain your reasoning.

Problem 224

What conclusion about the number 30 can be drawn from its factor pairs: 1, 2, 3, 5? A. Prime or composite? B. Prime? C. No prime factors? D. Composite?

Problem 225

Identify a prime factor of the composite number 58: A. 1, B. 7, C. 8, D. 29.

Problem 226

Express each number as powers in two different ways: a) 16 b) 81 c) 256 d) 1024 Examples: 424^{2}42, 242^{4}24 for 16.

Problem 227

Find the prime factorization of 500. Is it 22532^{2} 5^{3}2253, 22352^{2} 3^{5}2235, 22502^{2} 502250, or 5∗1005 * 1005∗100?

Problem 228

Find the prime factorization of 12. Options: 262^{6}26, 22⋅32^{2} \cdot 322⋅3, 242^{4}24, 8⋅88 \cdot 88⋅8, 2⋅62 \cdot 62⋅6.

Problem 229

Find the prime factorization of 18. Options: 595^{9}59, 2⋅322 \cdot 3^{2}2⋅32, 5⋅95 \cdot 95⋅9, 15315^{3}153.

Problem 230

Identify the factors of 21 from the list: 4, 12, 7, 3, 5, 2.

Problem 231

Why is there a coefficient of 3 in 543=323\sqrt[3]{54} = 3 \sqrt[3]{2}354​=332​? Consider the factors of 54.

Problem 232

Aus einer Urne mit 100 Kugeln: Bestimmen Sie die Ergebnisse und Wahrscheinlichkeiten für die Ereignisse A, B, C, D, E, F.

Problem 233

Identify which numbers are factors of 19,656, given its prime factorization 23×33×7×132^{3} \times 3^{3} \times 7 \times 1323×33×7×13. Options: 297, 16, 14.

Problem 234

Find the factors of 16,632 from these numbers based on their prime factorizations: 1. 81=3481=3^{4}81=342. 104=23×13104=2^{3} \times 13104=23×133. 33=3×1133=3 \times 1133=3×114. 28=22×728=2^{2} \times 728=22×75. 27=3327=3^{3}27=33

Problem 235

Find the prime factorization of 1683 in index form using the given prime factor tree.

Problem 236

Which of the following numbers are factors of 19,656 (23×33×7×132^{3} \times 3^{3} \times 7 \times 1323×33×7×13)? Select all that apply: 8=238=2^{3}8=23, 297=33×11297=3^{3} \times 11297=33×11, 63=32×763=3^{2} \times 763=32×7, 16=2416=2^{4}16=24, 14=2×714=2 \times 714=2×7.

Problem 237

Factor 132 into its prime components.

Problem 238

Find the possible row arrangements for 16 cheerleaders, with each row having the same number of members.

Problem 239

Find the smallest positive integer to multiply 360 by to make it a perfect square. Use prime factorization to explain.

Problem 240

Find the GCF of 28 and 35, then calculate $\frac{35}{1.35}$ . Finally, find the GCF of 40 and 240.

Determine the truth value of the converse: If $n$ is an odd natural number > 6, then $n$ is prime. Counterexample if false.

Find the missing number in each prime factorization: a) $200=2 \times 2 \times 1 \times 5 \times 5$ b) $216=2^{3} \times 3$ c) $8281=7 \times 7 \times 13 \times$ d) $1568=5 \times 7^{2}$

Find the smallest value of $\frac{p}{q}$ where $p$ is a common multiple of 4 and 9, and $q$ is a common factor of 72 and 120.

Express 600 as $2^{a} \times b \times c^{d}$ with $a, b, c, d$ as prime numbers. Determine $a, b, c, d$ .

Identify which of Dominic's factor pairs for 120 are both composite: A. $40 \times 3$ , B. $60 \times 2$ , C. $10 \times 12$ , D. $5 \times 24$ .

Express each number as powers in two different ways: a) 16 b) 81 c) 256 d) 1024 Examples: $4^{2}$ , $2^{4}$ for 16.

Find the prime factorization of 500. Is it $2^{2} 5^{3}$ , $2^{2} 3^{5}$ , $2^{2} 50$ , or $5 * 100$ ?

Find the prime factorization of 12. Options: $2^{6}$ , $2^{2} \cdot 3$ , $2^{4}$ , $8 \cdot 8$ , $2 \cdot 6$ .

Find the prime factorization of 18. Options: $5^{9}$ , $2 \cdot 3^{2}$ , $5 \cdot 9$ , $15^{3}$ .

Why is there a coefficient of 3 in $\sqrt[3]{54} = 3 \sqrt[3]{2}$ ? Consider the factors of 54.

Identify which numbers are factors of 19,656, given its prime factorization $2^{3} \times 3^{3} \times 7 \times 13$ . Options: 297, 16, 14.

Find the factors of 16,632 from these numbers based on their prime factorizations:
1. $81=3^{4}$
2. $104=2^{3} \times 13$
3. $33=3 \times 11$
4. $28=2^{2} \times 7$
5. $27=3^{3}$

Which of the following numbers are factors of 19,656 ( $2^{3} \times 3^{3} \times 7 \times 13$ )? Select all that apply: $8=2^{3}$ , $297=3^{3} \times 11$ , $63=3^{2} \times 7$ , $16=2^{4}$ , $14=2 \times 7$ .

If 77 divides $n$ , what are the other natural numbers that also divide $n$ ? List them.

Liste die Elemente der Mengen: a) $A=\{x \mid x$ ist eine Primzahl und $x > 3\}$ , b) $B=\{x \mid x$ ist eine Primzahl und $x < 20\}$ .

Which option shows the prime factorization of 200 in exponential form? $2^{2} \cdot 5^{2}$ , $8 \cdot 25$ , $2^{3} \cdot 5^{2}$ , or $2 \cdot 10^{2}$ ?

Respond to a student claiming $188_{\text{ten}}$ is $723_{\text{five}}$ by showing $7 \times 5^{2}+2 \times 5^{1}+3 \times 5^{0}=18$ .

1. Wandle 218 in Binär- und Hexadezimalzahl um.
2. Was ist die Zahl (B1.4) im Dezimalsystem?
3. Subtrahiere $b=01100101$ von $a=01001110$ im Zweierkomplement und gib das Ergebnis als Dezimalzahl an.

Find common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 1120 ( $2^{5} \times 5 \times 7$ ) from the options given.

Find the common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 1120 ( $2^{5} \times 5 \times 7$ ). Select all correct answers.

Find the HCF and LCM of 60 ( $2^2 \times 3 \times 5$ ) and 220 ( $2^2 \times 5 \times 11$ ).

Find the prime factorization of 60 and express it in index form: $60 = 2^a \times 3^b \times 5^c$ .

Find the highest common factor (HCF) of $A=2^{3} \times 3 \times 5$ and $B=2^{2} \times 3 \times 5^{2}$ .

Bestimme die Teilermengen für die folgenden Zahlen: $T_{36}$ , $T_{63}$ , $T_{56}$ , $T_{81}$ , $T_{100}$ , $T_{85}$ , $T_{37}$ , $T_{243}$ , $T_{225}$ .

Find the HCF of $G = 2^5 \times 3^3 \times 5$ and $H = 2^2 \times 3^6 \times 5 \times 7$ . Express in prime factor form.

Find the prime factor decomposition of 208 using $2^{3} \times 13$ for 104. Provide your answer in index form.