Number Theory

Problem 1301

How many divisors does the number 8256 have, given that Tesla car batteries contain 8,256 cells?

See Solution

Problem 1302

Find the residue of a number mod 60, given that it is 1 mod 3, 2 mod 4, and 3 mod 5.

See Solution

Problem 1303

Find the prime factorization of $80$ . Options: $2 \times 5 \times 7$ , $4 \times 4 \times 5$ , $2^{2} \times 5$ , $2^{4} \times 5$ .

See Solution

Problem 1304

Find a prime number between 30 and 50 where the tens digit is one more than the ones digit. What is the number?

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Problem 1305

Julie has 27 grapes for 3 friends. Explain if she can divide them evenly using factor pairs.

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Problem 1306

Find the prime factorization and exponential form of the number 70.

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Problem 1307

Find the maximum value of $\frac{a}{b}$ where $a$ is a factor of 72 and 120, and $b$ is a multiple of 6 and 9.

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Problem 1308

Prove that for integers $a, b$ , and $c \neq 0$ , $a \mid b$ if and only if $a c \mid b c$ .

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Problem 1309

What number has the prime factorization $2^{2} \cdot 3 \cdot 5^{2}$ ? Options: 990, 1,700, 300, 880.

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Problem 1310

Which of the following sets includes all divisors of 920: $2,3,4,6$ ; $2,3,6$ ; $2,4,5,10$ ; $2,3,5,6,9,10$ ?

See Solution

Problem 1311

Find the GCF of 42 and 36, and the LCM of 7 and 9 using the ladder method. GCF on the left, LCM in an L-shape.

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Problem 1312

Find the prime factorization of 44. What are the prime numbers that multiply to give 44?

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Problem 1313

Which pairs have a greatest common factor of 1: A) 16 and 27 B) 24 and 26 C) 25 and 30 D) 48 and 56?

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Problem 1314

What number has the prime factorization $2^{2} \cdot 3 \cdot 5^{2}$ ? (1 point) 990

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Problem 1315

What is the prime factorization of 81? a) $3^{4}$ b) $2 \cdot 3^{4}$ c) $3^{4} \cdot 5$ d) $3^{2} \cdot 9$

See Solution

Problem 1316

Find the prime factorization of 225. Options: $2 \cdot 3 \cdot 5^{2}$ , $3 \cdot 5^{2}$ , $3^{2} \cdot 5^{2}$ , $3^{2} \cdot 5$ .

See Solution

Problem 1317

Find the least common multiple (LCM) of 12 and 27. Options: 324, 108, 162, 27.

See Solution

Problem 1318

Express 70 as a product of its prime factors. $70 =$

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Problem 1319

Find the prime factors of 350.

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Problem 1320

Find the least common multiple (LCM) of these pairs: 25 and 10, 48 and ___, 8 and 18, 6 and 11.

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Problem 1321

Find the least common multiple of 8, 18, and $16$ .

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Problem 1322

Find the factors of 75. Determine if it is a prime or composite number.

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Problem 1323

Find the GCF of 35 and 28. Then use it to factor $35-28$ .
GCF = $\square$ $35-28 = \square \times (\square - \square)$

See Solution

Problem 1324

Colette has 24 math and 26 science books. What's the max number of bookshelves for even distribution with no leftovers?

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Problem 1325

Determine if the number 441 is prime or composite. If composite, provide its prime factorization.

See Solution

Problem 1326

Find two numbers under 20 with a GCF of 3 and an LCM of 60. What are the numbers?

See Solution

Problem 1327

Find the exponents in the prime factorization of 720: $2^{\square} \cdot 3^{\square} \cdot 5^{\square}$ . What are the values?

See Solution

Problem 1328

Find the prime factors of 140.

See Solution

Problem 1329

Lana has 78 pencils and 143 erasers. Find the GCD of 78 and 143 to determine the max identical boxes she can pack.

See Solution

Problem 1330

What odd number greater than 2 divides both 54 and 24?

See Solution

Problem 1331

Find the smallest number of students that can be divided by 6 and 8, but leaves a remainder when divided by 10. What is the remainder?

See Solution

Problem 1332

Find the number of outcomes where the sum of two dice is composite. There are $\square$ such outcomes.

See Solution

Problem 1333

What is the smallest number to multiply 1800 by to get a perfect cube?

See Solution

Problem 1334

Find the smallest positive integer $n$ for which $588n$ is a perfect square.

See Solution

Problem 1335

Is 9 a divisor of 47385?

See Solution

Problem 1336

Find the prime factorization of 440.

See Solution

Problem 1337

Is 2 a divisor of 3814? Options: 1) No, sum of digits is odd 2) Yes, sum of digits is odd 3) No, odd number 4) Yes, even number.

See Solution

Problem 1338

Is 9 a divisor of 867123?

See Solution

Problem 1339

Find the GCF of 54 and 30. Then factor $54 + 30$ using the GCF: $54 + 30 = \square \times (\square + \square)$

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Problem 1340

Find the GCF of 30 and 18. Then factor $30-18$ using the GCF: $30-18=\square \times(\square-\square)$ .

See Solution

Problem 1341

List all prime numbers between 30 and 55: A. $31,33,37,41,43,47,53$ B. $31,37,41,43,47,51,53$ C. $31,33,37,39,41,43,47,51,53$ D. $31,37,41,43,47,53$

See Solution

Problem 1342

Determine if the number 131 is prime, composite, or neither.

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Problem 1343

Find the GCF of 18 and 3. $\mathrm{GCF} =$

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Problem 1344

Find the row in the multiplication table where products alternate odd/even and the sum of the digits is 9.

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Problem 1345

Find two numbers that add to a prime number and multiply to a square number. What are these numbers?

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Problem 1346

Find the least common multiple of numbers with factorizations $2^{2} \cdot 3 \cdot 5$ and $2 \cdot 3^{2} \cdot 7$ .

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Problem 1347

Calculate the missing check digit for the UPC barcode: 83341900117X.

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Problem 1348

Identify the squares and their square roots: $2^2=4$ , $3^2=9$ , $4^2=16$ , $5^2=25$ . Find $\sqrt{4}$ , $\sqrt{9}$ , $\sqrt{16}$ , $\sqrt{25}$ .

See Solution

Problem 1349

Find the largest divisor of $13^{14} \times 31^{7} \times 57^{8}$ and $13^{5} \times 29^{18} \times 57^{5}$ .

See Solution

Problem 1350

Is it true that all odd numbers less than 15 are prime? Provide a counterexample.

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Problem 1351

Convert the number $21101.201_{(3)}$ from base 3 to base 10.

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Problem 1352

What is the maximum number of cookies per bag if Anna has 66 chocolate chip and 54 sugar cookies, with no leftovers? The answer is $\square$ .

See Solution

Problem 1353

Find the LCM of two numbers whose product is 2527 and GCD is 19. $\mathrm{LCM}=\square$

See Solution

Problem 1354

Find the GCD of 286 and 6,254 using the Euclidean algorithm. The GCD is $\square$ .

See Solution

Problem 1355

Find the largest four-digit number with exactly three positive factors. The answer is $\square$ .

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Problem 1356

Convert the numbers $(2110112.212)_{3}$ and $(3220)_{4}$ to decimal form.

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Problem 1357

Is the statement "There exists a real number that is not a whole number" true or false? If false, provide the negation.

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Problem 1358

Determine if 225 is prime or composite. If composite, provide its prime factorization.

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Problem 1359

How many students can sit in each row if there are 96 girls and 80 boys, with no mixing allowed? Find the GCD of 96 and 80.

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Problem 1360

Find the number of outcomes where the sum of two dice is composite. There are $\square$ such outcomes.

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Problem 1361

Which number is not a prime factor of 14014: (a) 7, (b) 11, (c) 13, or (d) 17?

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Problem 1362

Find the Greatest Common Factor (GCF) of 8 and 15 to determine how many balloons Finn will put in each bunch.

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Problem 1363

Find the greatest common factor of 6699 and 8265, given that $6699=3 \times 2233$ and $8265=3 \times 2755$ .

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Problem 1364

Find the value of $c$ in the prime factorization of the LCM of $33$ and $110$ : $2 \times 3 \times 5 \times 11$ .

See Solution

Problem 1365

Compare the numbers: $\pi^{3}$ , 27; 8, $\sqrt{75}$ ; $\frac{11}{7}$ , $\sqrt{2}$ ; $-\pi$ , -3 using < or >.

See Solution

Problem 1366

Is $24$ a divisor of $84.0$ ?

See Solution

Problem 1367

1) Find the largest prime factor of 12300. 2) Calculate the value of $\frac{1+\frac{1}{6}}{2-\frac{3}{4}}$ . 3) If Terry passed $80\%$ of his science tests and failed 4, how many did he pass?

See Solution

Problem 1368

Which number is a factor in the prime factorization of 60, which is $2^2 \times 3^1 \times 5^1$ ?

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Problem 1369

Find the highest common factor (HCF) of 75 and 105.

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Problem 1370

Find three prime numbers whose product is 1235.

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Problem 1371

Is 136 a perfect square? Explain why or why not using perfect squares and whole numbers.

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Problem 1372

Find two numbers, besides 64, that are both perfect squares and perfect cubes.

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Problem 1373

Find the least common multiple of 10 and 22. A. 2 B. 55 C. 220 D. 110

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Problem 1374

Is 9 a divisor of 974988?

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Problem 1375

Find the prime factors of 360.

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Problem 1376

Find the sum of the first 8 prime numbers.

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Problem 1377

Find the factors of 24.

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Problem 1378

Find the HCF of $A = 2^{2} \times 3 \times 5^{2}$ and $B = 2^{3} \times 5 \times 11$ using prime factors.

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Problem 1379

Find the HCF and LCM of $A=2^{2} \times 3 \times 5^{2}$ and $B=2^{3} \times 5 \times 11$ using prime factorization.

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Problem 1380

Find the HCF of $X = a^{3} \times b^{2} \times c^{2}$ and $Y = a^{5} \times b \times c \times d$ in terms of $a, b, c, d$ .

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Problem 1381

Find the HCF and LCM of $X=a^{3}b^{2}c^{2}$ and $Y=a^{5}bc d$ . HCF: $a^{3}bc$ , LCM: (answer in prime factors).

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Problem 1382

Which scoreboard shows prime final scores for both teams? A. HOME 45, VISITORS 19 B. HOME 31, VISITORS 49 C. HOME 7, VISITORS 47 D. HOME 13, VISITORS 33

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Problem 1383

List the prime numbers on a clock face: a. $1,3,5,7,11$ b. $3,5,7,9,11$ c. $2,3,7,9$ d. $2,3,5,7,11$

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Problem 1384

Is 33 a prime number? Circle all correct statements: a. Yes, all odd numbers are prime. b. Yes, it has 2 factors: 1 and 33. c. No, it's a multiple of 3. d. No, it has more than 2 factors: 1, 3, 11, 33. e. No, it's even.

See Solution

Problem 1385

Gigi grouped her dolls in 9s. Which sets contain multiples of 9? A. $1,9,15,27$ B. $9,18,27,45,63$ C. $9,19,29,39,49$ D. $9,18,27,36,44$

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Problem 1386

How many times greater is the value of the 2 in 270,413 compared to the 2 in 419,427?

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Problem 1387

Find the LCM of 273 ( $3 \times 7 \times 13$ ) and 858 ( $2 \times 3 \times 11 \times 13$ ) using their prime factorizations.

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Problem 1388

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

See Solution

Problem 1389

Draw the prime factor tree for 330 and find the highest common factor (HCF) of 308 and 330.

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Problem 1390

Find the LCM of 182 and 714 using their prime factorizations: $182 = 2 \times 7 \times 13$ , $714 = 2 \times 3 \times 7 \times 17$ .

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Problem 1391

Find the highest common factor (HCF) of 198 and 330 using their prime factors: $198 = 2 \times 3^2 \times 11$ , $330 = 2 \times 3 \times 5 \times 11$ .

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Problem 1392

Find the lowest common multiple (LCM) of 105 and 130 using their prime factors. Prime factors of 105 are $3, 5, 7$ .

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Problem 1393

There are no square numbers that are also prime numbers. Explain why this is the case.

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Problem 1394

Find the highest common factor (HCF) of 495 and 522 using their prime factors: $495 = 3^2 \times 5 \times 11$ , $522 = 2 \times 3^2 \times 29$ .

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Problem 1395

Write $N = 20^{6} \times 10^{2}$ as a product of prime factors in index form.

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Problem 1396

Mina baked 50 muffins. How can she arrange them equally on plates? List the factor pairs of 50.

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Problem 1397

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

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Problem 1398

Prove by induction that for $n \geq 1$ , $42$ divides $(n^{7} - n)$ .

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Problem 1399

Identify the number that is not a factor of 75: 75, 4, 25, or 3.

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Problem 1400

What is the prime factorization of 32? Choose from: $2^{2} \cdot 3^{2} \cdot 5$ , $2^{2}+3+5+7+11$ , $2^{4}+4^{2}$ , $2^{5}$ .

See Solution

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Number Theory

Problem 1301

How many divisors does the number 8256 have, given that Tesla car batteries contain 8,256 cells?

Problem 1302

Find the residue of a number mod 60, given that it is 1 mod 3, 2 mod 4, and 3 mod 5.

Problem 1303

Find the prime factorization of 808080. Options: 2×5×72 \times 5 \times 72×5×7, 4×4×54 \times 4 \times 54×4×5, 22×52^{2} \times 522×5, 24×52^{4} \times 524×5.

Problem 1304

Find a prime number between 30 and 50 where the tens digit is one more than the ones digit. What is the number?

Problem 1305

Julie has 27 grapes for 3 friends. Explain if she can divide them evenly using factor pairs.

Problem 1306

Find the prime factorization and exponential form of the number 70.

Problem 1307

Find the maximum value of ab\frac{a}{b}ba​ where aaa is a factor of 72 and 120, and bbb is a multiple of 6 and 9.

Problem 1308

Prove that for integers a,ba, ba,b, and c≠0c \neq 0c=0, a∣ba \mid ba∣b if and only if ac∣bca c \mid b cac∣bc.

Problem 1309

What number has the prime factorization 22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}22⋅3⋅52? Options: 990, 1,700, 300, 880.

Problem 1310

Which of the following sets includes all divisors of 920: 2,3,4,62,3,4,62,3,4,6; 2,3,62,3,62,3,6; 2,4,5,102,4,5,102,4,5,10; 2,3,5,6,9,102,3,5,6,9,102,3,5,6,9,10?

Problem 1311

Find the GCF of 42 and 36, and the LCM of 7 and 9 using the ladder method. GCF on the left, LCM in an L-shape.

Problem 1312

Find the prime factorization of 44. What are the prime numbers that multiply to give 44?

Problem 1313

Which pairs have a greatest common factor of 1: A) 16 and 27 B) 24 and 26 C) 25 and 30 D) 48 and 56?

Problem 1314

What number has the prime factorization 22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}22⋅3⋅52? (1 point) 990

Problem 1315

What is the prime factorization of 81? a) 343^{4}34 b) 2⋅342 \cdot 3^{4}2⋅34 c) 34⋅53^{4} \cdot 534⋅5 d) 32⋅93^{2} \cdot 932⋅9

Problem 1316

Find the prime factorization of 225. Options: 2⋅3⋅522 \cdot 3 \cdot 5^{2}2⋅3⋅52, 3⋅523 \cdot 5^{2}3⋅52, 32⋅523^{2} \cdot 5^{2}32⋅52, 32⋅53^{2} \cdot 532⋅5.

Problem 1317

Find the least common multiple (LCM) of 12 and 27. Options: 324, 108, 162, 27.

Problem 1318

Express 70 as a product of its prime factors. 70= 70 = 70=

Problem 1319

Find the prime factors of 350.

Problem 1320

Find the least common multiple (LCM) of these pairs: 25 and 10, 48 and ___, 8 and 18, 6 and 11.

Problem 1321

Find the least common multiple of 8, 18, and 161616.

Problem 1322

Find the factors of 75. Determine if it is a prime or composite number.

Problem 1323

Find the GCF of 35 and 28. Then use it to factor 35−2835-2835−28. GCF = □\square□ 35−28=□×(□−□)35-28 = \square \times (\square - \square)35−28=□×(□−□)

Problem 1324

Colette has 24 math and 26 science books. What's the max number of bookshelves for even distribution with no leftovers?

Problem 1325

Determine if the number 441 is prime or composite. If composite, provide its prime factorization.

Problem 1326

Find two numbers under 20 with a GCF of 3 and an LCM of 60. What are the numbers?

Problem 1327

Find the exponents in the prime factorization of 720: 2□⋅3□⋅5□2^{\square} \cdot 3^{\square} \cdot 5^{\square}2□⋅3□⋅5□. What are the values?

Problem 1328

Find the prime factors of 140.

Problem 1329

Lana has 78 pencils and 143 erasers. Find the GCD of 78 and 143 to determine the max identical boxes she can pack.

Problem 1330

What odd number greater than 2 divides both 54 and 24?

Problem 1331

Find the smallest number of students that can be divided by 6 and 8, but leaves a remainder when divided by 10. What is the remainder?

Problem 1332

Find the number of outcomes where the sum of two dice is composite. There are □\square□ such outcomes.

Problem 1333

What is the smallest number to multiply 1800 by to get a perfect cube?

Problem 1334

Find the smallest positive integer nnn for which 588n588n588n is a perfect square.

Problem 1335

Is 9 a divisor of 47385?

Problem 1336

Find the prime factorization of 440.

Problem 1337

Is 2 a divisor of 3814? Options: 1) No, sum of digits is odd 2) Yes, sum of digits is odd 3) No, odd number 4) Yes, even number.

Problem 1338

Is 9 a divisor of 867123?

Problem 1339

Find the GCF of 54 and 30. Then factor 54+3054 + 3054+30 using the GCF: 54+30=□×(□+□)54 + 30 = \square \times (\square + \square)54+30=□×(□+□)

Problem 1340

Find the prime factorization of $80$ . Options: $2 \times 5 \times 7$ , $4 \times 4 \times 5$ , $2^{2} \times 5$ , $2^{4} \times 5$ .

Find the maximum value of $\frac{a}{b}$ where $a$ is a factor of 72 and 120, and $b$ is a multiple of 6 and 9.

Prove that for integers $a, b$ , and $c \neq 0$ , $a \mid b$ if and only if $a c \mid b c$ .

What number has the prime factorization $2^{2} \cdot 3 \cdot 5^{2}$ ? Options: 990, 1,700, 300, 880.

Which of the following sets includes all divisors of 920: $2,3,4,6$ ; $2,3,6$ ; $2,4,5,10$ ; $2,3,5,6,9,10$ ?

What number has the prime factorization $2^{2} \cdot 3 \cdot 5^{2}$ ? (1 point) 990

What is the prime factorization of 81? a) $3^{4}$ b) $2 \cdot 3^{4}$ c) $3^{4} \cdot 5$ d) $3^{2} \cdot 9$

Find the prime factorization of 225. Options: $2 \cdot 3 \cdot 5^{2}$ , $3 \cdot 5^{2}$ , $3^{2} \cdot 5^{2}$ , $3^{2} \cdot 5$ .

Express 70 as a product of its prime factors. $70 =$

Find the least common multiple of 8, 18, and $16$ .

Find the GCF of 35 and 28. Then use it to factor $35-28$ .
GCF = $\square$ $35-28 = \square \times (\square - \square)$

Find the exponents in the prime factorization of 720: $2^{\square} \cdot 3^{\square} \cdot 5^{\square}$ . What are the values?

Find the number of outcomes where the sum of two dice is composite. There are $\square$ such outcomes.

Find the smallest positive integer $n$ for which $588n$ is a perfect square.

Find the GCF of 54 and 30. Then factor $54 + 30$ using the GCF: $54 + 30 = \square \times (\square + \square)$

Find the GCF of 30 and 18. Then factor $30-18$ using the GCF: $30-18=\square \times(\square-\square)$ .

Find the GCF of 18 and 3. $\mathrm{GCF} =$

Find the least common multiple of numbers with factorizations $2^{2} \cdot 3 \cdot 5$ and $2 \cdot 3^{2} \cdot 7$ .

Identify the squares and their square roots: $2^2=4$ , $3^2=9$ , $4^2=16$ , $5^2=25$ . Find $\sqrt{4}$ , $\sqrt{9}$ , $\sqrt{16}$ , $\sqrt{25}$ .

Find the largest divisor of $13^{14} \times 31^{7} \times 57^{8}$ and $13^{5} \times 29^{18} \times 57^{5}$ .

Convert the number $21101.201_{(3)}$ from base 3 to base 10.

What is the maximum number of cookies per bag if Anna has 66 chocolate chip and 54 sugar cookies, with no leftovers? The answer is $\square$ .

Find the LCM of two numbers whose product is 2527 and GCD is 19. $\mathrm{LCM}=\square$

Find the GCD of 286 and 6,254 using the Euclidean algorithm. The GCD is $\square$ .

Find the largest four-digit number with exactly three positive factors. The answer is $\square$ .

Convert the numbers $(2110112.212)_{3}$ and $(3220)_{4}$ to decimal form.

Find the number of outcomes where the sum of two dice is composite. There are $\square$ such outcomes.

Find the greatest common factor of 6699 and 8265, given that $6699=3 \times 2233$ and $8265=3 \times 2755$ .

Find the value of $c$ in the prime factorization of the LCM of $33$ and $110$ : $2 \times 3 \times 5 \times 11$ .

Compare the numbers: $\pi^{3}$ , 27; 8, $\sqrt{75}$ ; $\frac{11}{7}$ , $\sqrt{2}$ ; $-\pi$ , -3 using < or >.

Is $24$ a divisor of $84.0$ ?

1) Find the largest prime factor of 12300. 2) Calculate the value of $\frac{1+\frac{1}{6}}{2-\frac{3}{4}}$ . 3) If Terry passed $80\%$ of his science tests and failed 4, how many did he pass?

Which number is a factor in the prime factorization of 60, which is $2^2 \times 3^1 \times 5^1$ ?