Number Theory

Problem 1201

Determine if the following statements are true or false, and explain why: a. If a number has 8 as a factor, it has 2 as a factor. b. If a number has 5 as a factor, it has 10 as a factor.

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Problem 1202

Determine if these statements are true or false, and explain why: a. If a number has 8 as a factor, it has 2 as a factor. b. If a number has 5 as a factor, it has 10 as a factor.

See Solution

Problem 1203

If 15 divides $n$ , what other natural numbers divide $n$ and why? List them and explain.

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Problem 1204

Find the least common multiple (LCM) of 5 and 4: $LCM(5, 4)$ .

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Problem 1205

What is the probability of selecting a prime number from the set $N=\{2,3,5,7,9\}$ ?

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Problem 1206

Find the prime factorization for 20 and 35. Match them with the correct letter from the options: (U), (B), (E), (G), (H).

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Problem 1207

Ist der Ausdruck $\frac{n \cdot(n+1) \cdot(2 n+1)}{6}$ für alle $n \geq 1$ ganzzahlig? Begründen Sie!

See Solution

Problem 1208

Find the largest prime factor of $12300$ .

See Solution

Problem 1209

Find the check digit $d$ for the UPC code $0-34000-12080-d$ using the standard UPC check digit formula.

See Solution

Problem 1210

Find the prime factorization of 54.

See Solution

Problem 1211

Express 52 as a product of its prime factors.

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Problem 1212

Welche Rechnungen sind falsch? Begründe mit der letzten Ziffer oder der Anzahl der Stellen. a) $7,934^{2}=62,9486$ b) $25,63^{2}=656,8969$ c) $20,563^{2}=422,836963$ d) $560^{2}=3136000$ e) $142^{2}=201602$

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Problem 1213

Find the least number of children in a football club that leaves a remainder of 2 when divided by both 9 and 12.

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Problem 1214

Is -6 a rational number? True or False?

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Problem 1215

Complete the prime factorization of 84 using the numbers: 2, 3, 5, 7. Fill in the boxes: $84 = \square \times 12$ , $= \square \times 4 \times 7$ , $= \square \times \square \times 3 \times 7$ .

See Solution

Problem 1216

Determine if 47 is prime or find its prime factorization.
A. The prime factorization is $47=\square$ . B. The number is prime.

See Solution

Problem 1217

Identify the composite number between 1 and 10: A. 5, B. 8, C. 3, D. 2.

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Problem 1218

Is $5 \div 0.4$ rational or irrational? Choose the correct reason: a, b, c, or d.

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Problem 1219

Find a number between 62 and 67 with prime factors 2 and 11. What is the number?

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Problem 1220

Find the greatest common factor (GCF) of 16 in. and 28 in. to determine the largest tile size Kennedy can use.

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Problem 1221

Factor 300 into prime factors and show your steps clearly.

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Problem 1222

Select all the prime numbers from the hundreds chart: 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50.

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Problem 1223

Identify the prime number between 10 and 20 from these options: A. 18, B. 12, C. 15, D. 19.

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Problem 1224

Identify the prime number between 60 and 70: A. 63, B. 65, C. 67, D. 69.

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Problem 1225

Identify the prime number between 40 and $50$ : A. 43, B. 45, C. 49, D. 46.

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Problem 1226

Identify the composite number between 1 and 10 from these options: A. 9 B. 7 C. 5 D. 3.

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Problem 1227

How many sailboats, a multiple of 2 and 7, are on the lake? A. 21 B. 28 C. 35 D. 38

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Problem 1228

Check all the prime numbers from this list: 2, 3, 5, 8, 11, 27, None of the above.

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Problem 1229

Identify TWO prime numbers from the list: 91, 87, 43, 57, 49, 37. Provide proof for your selections.

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Problem 1230

Identify TWO prime numbers from: 91, 87, 43, 57, 49, 37. Provide proof for your selections.

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Problem 1231

Find the prime factorizations for these numbers: 70, 121, 150, 40, 180, 240, 60, 66, 95, 52.

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Problem 1232

List all the prime numbers between 10 and 20.

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Problem 1233

How many whole number factors does 64 have? Is it a perfect square? Options: 7, 8, 5, 6.

See Solution

Problem 1234

What is the prime factorization of 20? Options: $3 \times 3 \times 5$ , $2 \times 2 \times 5$ , $2 \times 5 \times 5$ , $2 \times 3 \times 5$ .

See Solution

Problem 1235

Find the prime factorization of 700.

See Solution

Problem 1236

Ist der Ausdruck $\frac{n \cdot(n+1) \cdot(2 n+1)}{6}$ für alle $n \geq 1$ ganzzahlig? Begründen Sie!

See Solution

Problem 1237

What type of number can't be expressed as a fraction $\frac{a}{b}$ with integers $a$ , $b$ ( $b \neq 0$ )?

See Solution

Problem 1238

Find the prime factors of 84.

See Solution

Problem 1239

Find the least common multiple of 30 and $50$ . A) 10 B) 75 C) 750 D) 150

See Solution

Problem 1240

Find the prime factors of $90$ . (A) 1, 90 (B) 1, 3 (C) $2, 3, 5, 90$ (D) $2, 3, 5$

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Problem 1241

Identify prime and composite numbers from 1 to 30. What are the prime factorizations of 30 and 24?

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Problem 1242

Find the GCF of the numbers $16$ , $44$ , and $72$ .

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Problem 1243

List the factors of the numbers: 12, 42, and 60.

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Problem 1244

Find the prime factorization of 213, which is calculated as $3^{2} \times 11$ .

See Solution

Problem 1245

Find the number of days $x$ when both medications are given together, with one every 4 days and the other every 6 days.

See Solution

Problem 1246

Find the probability that a randomly chosen three-digit number from $\{0,1,2,3,4,5\}$ is not a multiple of 5.

See Solution

Problem 1247

Find the least common multiple of 5, 11, and 45 to determine when planets A, B, and C align again.

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Problem 1248

Find the prime factorization of 3993 in the form $p \times q$ and calculate $p \times q \times r$ . Choose from: 99, 63, 875, 105.

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Problem 1249

Find the prime factorization of 196 and explain how to use it to calculate $\sqrt{196}$ .

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Problem 1250

Find the divisors of $a=15$ and $b=25$ , list common ones, and determine the GCD of $a$ and $b$ .

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Problem 1251

Find the prime factorization of 44 using multiplication.

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Problem 1252

Find the prime factorization of 71. If it's prime, just state the number.

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Problem 1253

Find the least common multiple (LCM) of 5 and 12. Options: 17, 60, 30, 36.

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Problem 1254

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

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Problem 1255

Identify the correct terms: Numbers like $1, 4, 9, 25$ are perfect squares, while $1, 8, 27, 125$ are perfect cubes.

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Problem 1256

Which statement about 16 is true? (A) Divisible by 4 numbers. (B) All factors are composite. (C) All multiples are factors. (D) 16 is a multiple of its factors.

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Problem 1257

Rachel's locker number is prime. Which of these could be her locker number: 9, 18, 27, 11, 23, 31?

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Problem 1258

Identify which of the numbers 190, 191, 865 are divisible by 5, 2, and 10. Mark all that apply.

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Problem 1259

Find the prime factorization of 33 and list the factors in ascending order (e.g., $2 \times 2 \times 3 \times 5$ ).

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Problem 1260

Find the prime factorization of 5 and list the factors in ascending order (e.g., $2 \times 2 \times 3 \times 5$ ).

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Problem 1261

Find the prime factorization of 80 and list the factors from least to greatest (e.g., $2 \times 2 \times 2 \times 5$ ).

See Solution

Problem 1262

Find the prime factorization of 64 and list the factors in ascending order, e.g., $2 \times 2 \times 2 \times 2 \times 2 \times 2$ .

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Problem 1263

Find the prime factorization of 27 and list the factors from least to greatest, like $2 \times 2 \times 3 \times 5$ .

See Solution

Problem 1264

Find the prime factorization of 200, listing factors from least to greatest, like $2 \times 2 \times 5$ .

See Solution

Problem 1265

Find the prime factorization of 112 and list the factors from least to greatest, like $2 \times 2 \times 3 \times 5$ .

See Solution

Problem 1266

Does an irrational number plus a rational number equal an irrational number? Prove or provide counterexamples.

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Problem 1267

Define prime and composite numbers: A composite number has more than 2 factors, while a prime number has exactly 2 factors.

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Problem 1268

Find the Least Common Multiple (LCM) of 12 and 7.

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Problem 1269

Find pairs of numbers with a greatest common factor of 6: A. 18 and 24, B. 21 and 36, C. 30 and 48, D. 36 and 60, E. 48 and 72, F. 54 and 84.

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Problem 1270

Identify the irrational number from the options: A $\frac{6}{7}$ , B 18.7, C 3.14, D $\sqrt{47}$ .

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Problem 1271

List 3 multiples and 3 factors of 24.

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Problem 1272

Harley has 540 football and 486 baseball cards. What is the largest stack size for equal card types? Answer: $\square$ .

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Problem 1273

Check if 196 is a square number.

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Problem 1274

Is 256 a square number? Determine if there exists an integer $n$ such that $n^2 = 256$ .

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Problem 1275

Identify the prime factorization of 72 in exponential form: $2^{3} \cdot 3^{2}$ , $8 \cdot 9$ , $2^{2} \cdot 3^{2}$ , or $2 \cdot 6^{2}$ ?

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Problem 1276

What is the largest number of cards per stack if Harley has 1500 football cards and 486 baseball cards?

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Problem 1277

Find the least common multiple of 30, 40, and $70$ . A. 84 B. 120 C. 840 D. 1,200 E. 84,000

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Problem 1278

Find the LCM of $x=5 \times 7^{4} \times 11^{4} \times 13^{4}$ and $y=2^{3} \times 7^{2} \times 11^{3} \times 13^{3}$ . Answer in index form.

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Problem 1279

Find the HCF of $p=2^{2} \times 7^{4} \times 11^{5} \times 13^{4}$ and $q=2^{3} \times 3^{2} \times 7 \times 13^{2}$ in index form.

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Problem 1280

Find the greatest number of equal containers Mr. Pinello can make with 80 colored pencils and 64 markers. Show your work.

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Problem 1281

Identify the true statements:
1. 33 has more than two factors.
2. All factors of 34 are even.
3. 35 has two factors.
4. 3 is a factor of 36.
5. 37 is prime.
6. 38 is composite.

See Solution

Problem 1282

Check which numbers are prime: 7, 11, 12, 13, 23, 27, or "None of the above".

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Problem 1283

Find the largest number of identical teams that can be formed from 312 cadets and 364 junior officers.

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Problem 1284

Kennedy's wall is 16 in. high and 28 in. long. What is the side length of the largest square tile she can use?

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Problem 1285

Find the greatest common divisor (GCD) of 24 and 40 to determine how many equal packages of paintbrushes and paint can be made.

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Problem 1286

Find the check digit $d$ for the Visa Card number 416200123456789d.

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Problem 1287

Is the number 7 prime? Answer YES or NO.

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Problem 1288

Is 12 a composite number? Answer YES or NO.

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Problem 1289

Find all factors of 12 and the highest common factor (HCF) of 18 and 12. Factors of 18: $1,2,3,6,9,18$ .

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Problem 1290

List the first six multiples of 3 and 4. What is the lowest common multiple (LCM) of 3 and 4?

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Problem 1291

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

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Problem 1292

What does the "shrinking algorithm" indicate about the GCF of 42 and 180?

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Problem 1293

Find the prime factorization of 990.

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Problem 1294

Find the prime factorization of 25.

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Problem 1295

Find the prime factorization of 10.

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Problem 1296

Find the prime factorization of 61. If it's prime, just state the number.

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Problem 1297

Find the GCF and LCM of A, B, and C. Express answers as products of prime powers in increasing order.
A = 3 \cdot 5 \cdot 19^{7}, B = 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 13^{9} \cdot 17, C = 2^{5} \cdot 5^{6} \cdot 7^{2} \cdot 11 \cdot 17^{3}.

See Solution

Problem 1298

Find the prime factorization of 1701 using exponents for repeated factors: 1701 = $\square$

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Problem 1299

Factor 66 into its prime factors. $66 =$

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Problem 1300

Carmen listed the multiples of 24 as 1, $2, 3, 4, 6, 8, 12$ , and 24. Is this correct? Explain why. Also, what is the least common multiple of 6 and 8?

See Solution

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Number Theory

Problem 1201

Determine if the following statements are true or false, and explain why: a. If a number has 8 as a factor, it has 2 as a factor. b. If a number has 5 as a factor, it has 10 as a factor.

Problem 1202

Determine if these statements are true or false, and explain why: a. If a number has 8 as a factor, it has 2 as a factor. b. If a number has 5 as a factor, it has 10 as a factor.

Problem 1203

If 15 divides nnn, what other natural numbers divide nnn and why? List them and explain.

Problem 1204

Find the least common multiple (LCM) of 5 and 4: LCM(5,4)LCM(5, 4)LCM(5,4).

Problem 1205

What is the probability of selecting a prime number from the set N={2,3,5,7,9}N=\{2,3,5,7,9\}N={2,3,5,7,9}?

Problem 1206

Find the prime factorization for 20 and 35. Match them with the correct letter from the options: (U), (B), (E), (G), (H).

Problem 1207

Ist der Ausdruck n⋅(n+1)⋅(2n+1)6\frac{n \cdot(n+1) \cdot(2 n+1)}{6}6n⋅(n+1)⋅(2n+1)​ für alle n≥1n \geq 1n≥1 ganzzahlig? Begründen Sie!

Problem 1208

Find the largest prime factor of 123001230012300.

Problem 1209

Find the check digit ddd for the UPC code 0−34000−12080−d0-34000-12080-d0−34000−12080−d using the standard UPC check digit formula.

Problem 1210

Find the prime factorization of 54.

Problem 1211

Express 52 as a product of its prime factors.

Problem 1212

Problem 1213

Find the least number of children in a football club that leaves a remainder of 2 when divided by both 9 and 12.

Problem 1214

Is -6 a rational number? True or False?

Problem 1215

Complete the prime factorization of 84 using the numbers: 2, 3, 5, 7. Fill in the boxes: 84=□×1284 = \square \times 1284=□×12, =□×4×7= \square \times 4 \times 7=□×4×7, =□×□×3×7= \square \times \square \times 3 \times 7=□×□×3×7.

Problem 1216

Determine if 47 is prime or find its prime factorization. A. The prime factorization is 47=□47=\square47=□. B. The number is prime.

Problem 1217

Identify the composite number between 1 and 10: A. 5, B. 8, C. 3, D. 2.

Problem 1218

Is 5÷0.45 \div 0.45÷0.4 rational or irrational? Choose the correct reason: a, b, c, or d.

Problem 1219

Find a number between 62 and 67 with prime factors 2 and 11. What is the number?

Problem 1220

Find the greatest common factor (GCF) of 16 in. and 28 in. to determine the largest tile size Kennedy can use.

Problem 1221

Factor 300 into prime factors and show your steps clearly.

Problem 1222

Select all the prime numbers from the hundreds chart: 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50.

Problem 1223

Identify the prime number between 10 and 20 from these options: A. 18, B. 12, C. 15, D. 19.

Problem 1224

Identify the prime number between 60 and 70: A. 63, B. 65, C. 67, D. 69.

Problem 1225

Identify the prime number between 40 and 505050: A. 43, B. 45, C. 49, D. 46.

Problem 1226

Identify the composite number between 1 and 10 from these options: A. 9 B. 7 C. 5 D. 3.

Problem 1227

How many sailboats, a multiple of 2 and 7, are on the lake? A. 21 B. 28 C. 35 D. 38

Problem 1228

Check all the prime numbers from this list: 2, 3, 5, 8, 11, 27, None of the above.

Problem 1229

Identify TWO prime numbers from the list: 91, 87, 43, 57, 49, 37. Provide proof for your selections.

Problem 1230

Identify TWO prime numbers from: 91, 87, 43, 57, 49, 37. Provide proof for your selections.

Problem 1231

Find the prime factorizations for these numbers: 70, 121, 150, 40, 180, 240, 60, 66, 95, 52.

Problem 1232

List all the prime numbers between 10 and 20.

Problem 1233

How many whole number factors does 64 have? Is it a perfect square? Options: 7, 8, 5, 6.

Problem 1234

What is the prime factorization of 20? Options: 3×3×53 \times 3 \times 53×3×5, 2×2×52 \times 2 \times 52×2×5, 2×5×52 \times 5 \times 52×5×5, 2×3×52 \times 3 \times 52×3×5.

Problem 1235

Find the prime factorization of 700.

Problem 1236

Ist der Ausdruck n⋅(n+1)⋅(2n+1)6\frac{n \cdot(n+1) \cdot(2 n+1)}{6}6n⋅(n+1)⋅(2n+1)​ für alle n≥1n \geq 1n≥1 ganzzahlig? Begründen Sie!

Problem 1237

What type of number can't be expressed as a fraction ab\frac{a}{b}ba​ with integers aaa, bbb (b≠0b \neq 0b=0)?

Problem 1238

Find the prime factors of 84.

Problem 1239

Find the least common multiple of 30 and 505050. A) 10 B) 75 C) 750 D) 150

Problem 1240

Find the prime factors of 909090. (A) 1, 90 (B) 1, 3 (C) 2,3,5,902, 3, 5, 902,3,5,90 (D) 2,3,52, 3, 52,3,5

If 15 divides $n$ , what other natural numbers divide $n$ and why? List them and explain.

Find the least common multiple (LCM) of 5 and 4: $LCM(5, 4)$ .

What is the probability of selecting a prime number from the set $N=\{2,3,5,7,9\}$ ?

Ist der Ausdruck $\frac{n \cdot(n+1) \cdot(2 n+1)}{6}$ für alle $n \geq 1$ ganzzahlig? Begründen Sie!

Find the largest prime factor of $12300$ .

Find the check digit $d$ for the UPC code $0-34000-12080-d$ using the standard UPC check digit formula.

Complete the prime factorization of 84 using the numbers: 2, 3, 5, 7. Fill in the boxes: $84 = \square \times 12$ , $= \square \times 4 \times 7$ , $= \square \times \square \times 3 \times 7$ .

Determine if 47 is prime or find its prime factorization.
A. The prime factorization is $47=\square$ . B. The number is prime.

Is $5 \div 0.4$ rational or irrational? Choose the correct reason: a, b, c, or d.

Identify the prime number between 40 and $50$ : A. 43, B. 45, C. 49, D. 46.

What is the prime factorization of 20? Options: $3 \times 3 \times 5$ , $2 \times 2 \times 5$ , $2 \times 5 \times 5$ , $2 \times 3 \times 5$ .

Ist der Ausdruck $\frac{n \cdot(n+1) \cdot(2 n+1)}{6}$ für alle $n \geq 1$ ganzzahlig? Begründen Sie!

What type of number can't be expressed as a fraction $\frac{a}{b}$ with integers $a$ , $b$ ( $b \neq 0$ )?

Find the least common multiple of 30 and $50$ . A) 10 B) 75 C) 750 D) 150

Find the prime factors of $90$ . (A) 1, 90 (B) 1, 3 (C) $2, 3, 5, 90$ (D) $2, 3, 5$

Find the GCF of the numbers $16$ , $44$ , and $72$ .

Find the prime factorization of 213, which is calculated as $3^{2} \times 11$ .

Find the number of days $x$ when both medications are given together, with one every 4 days and the other every 6 days.

Find the probability that a randomly chosen three-digit number from $\{0,1,2,3,4,5\}$ is not a multiple of 5.

Find the prime factorization of 3993 in the form $p \times q$ and calculate $p \times q \times r$ . Choose from: 99, 63, 875, 105.

Find the prime factorization of 196 and explain how to use it to calculate $\sqrt{196}$ .

Find the divisors of $a=15$ and $b=25$ , list common ones, and determine the GCD of $a$ and $b$ .

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

Identify the correct terms: Numbers like $1, 4, 9, 25$ are perfect squares, while $1, 8, 27, 125$ are perfect cubes.

Find the prime factorization of 33 and list the factors in ascending order (e.g., $2 \times 2 \times 3 \times 5$ ).

Find the prime factorization of 5 and list the factors in ascending order (e.g., $2 \times 2 \times 3 \times 5$ ).

Find the prime factorization of 80 and list the factors from least to greatest (e.g., $2 \times 2 \times 2 \times 5$ ).

Find the prime factorization of 64 and list the factors in ascending order, e.g., $2 \times 2 \times 2 \times 2 \times 2 \times 2$ .

Find the prime factorization of 27 and list the factors from least to greatest, like $2 \times 2 \times 3 \times 5$ .

Find the prime factorization of 200, listing factors from least to greatest, like $2 \times 2 \times 5$ .

Find the prime factorization of 112 and list the factors from least to greatest, like $2 \times 2 \times 3 \times 5$ .

Identify the irrational number from the options: A $\frac{6}{7}$ , B 18.7, C 3.14, D $\sqrt{47}$ .

Harley has 540 football and 486 baseball cards. What is the largest stack size for equal card types? Answer: $\square$ .

Is 256 a square number? Determine if there exists an integer $n$ such that $n^2 = 256$ .