Number Theory

Problem 301

Show that if $p, q \in \mathbb{N}$ and $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\cdots+\frac{1}{1319}$ , then $p$ is divisible by 1979.

See Solution

Problem 302

Draw the prime factor tree for 190 and find the LCM of 105 and 190.

See Solution

Problem 303

Find which of the following numbers are factors of 16,632, given its prime factorization: $2^{3} \times 3^{3} \times 7 \times 11$ . Check these:
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 304

Find the prime factorization of 396 in index form and identify which of the following numbers are factors: 88, 14, 9, 121, 22.

See Solution

Problem 305

Find the prime factorization of 192 and list the factors in ascending order.

See Solution

Problem 306

Express 80 as a product of prime factors in ascending order.

See Solution

Problem 307

Find five pairs of prime numbers that add up to 48.

See Solution

Problem 308

Identify which of the following numbers are factors of 28,728, given its prime factorization: $2^{3} \times 3^{3} \times 7 \times 19$ .
1. $63=3^{2} \times 7$
2. $16=2^{4}$

See Solution

Problem 309

The LCM of two numbers has two $2$ s and three $3$ s. What can you deduce about these numbers?

See Solution

Problem 310

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

See Solution

Problem 311

Find the minimum 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 312

Amiyah landed on a prime number between 1 and 50. Which number could NOT be it: 43, 31, 21, or 13?

See Solution

Problem 313

Find the least common multiple of $A=3 \cdot 5 \cdot 17 \cdot 37$ and $B=3^3 \cdot 5 \cdot 11^3 \cdot 37$ .

See Solution

Problem 314

How many digits does the product of $2^{82589933}-1$ and 11 have beyond 24862048 digits?

See Solution

Problem 315

Find all numbers that are multiples of 30.

See Solution

Problem 316

List all the factors of $81$ .

See Solution

Problem 317

Circle the composite numbers from this list: 15, 2, 16, 17, 29, 27, 8, 10, 0, 9, 6, 11, 3, 20. Composite numbers have more than 2 factors.

See Solution

Problem 318

Find the factors of $m = 2^{5} \times x \times y^{3}$ , where $x$ and $y$ are primes > 2. Select all that apply: $x^{2}, 8, xy, 2y^{4}, 4y^{2}, 64x$ .

See Solution

Problem 319

Find the prime factorization of 240. Also, determine if the factorization of 90 is prime.

See Solution

Problem 320

Find the prime factorization of 240. Use the numbers: $2^{\wedge} 4$ , $3^{\wedge} 1$ , $5^{\wedge} 1$ .

See Solution

Problem 321

Is the prime factorization of 90 correct? Choose from: $5 \cdot 2 \cdot 9$ , $2 \cdot 3 \cdot 15$ , or $2 \cdot 3 \cdot 3 \cdot 5$ .

See Solution

Problem 322

Identify the relatively prime pair from these options: A) 72 and 115 B) 120 and 835 C) 201 and 735 D) 303 and 1041.

See Solution

Problem 323

Find the greatest length in meters for equal pieces from 63 m of pink and 105 m of green ribbon with no leftovers.

See Solution

Problem 324

Find the GCD of 30 and 42, and show it as a product of its prime factors: $30 = 2 \cdot 3 \cdot 5$ , $42 = 2 \cdot 3 \cdot 7$ .

See Solution

Problem 325

a) Count the prime numbers that are multiples of 5. b) Count the prime numbers that are multiples of 8.

See Solution

Problem 326

Find the common factors of 56 and 28. What is the value of $\frac{\frac{56}{1 \times 56}}{-x}+\frac{\frac{28}{1 \times 28}}{-x}$ ?

See Solution

Problem 327

Determine if these statements are always, sometimes, or never true: (a) The square of a non-zero rational number is negative. (b) The product of two negative rational numbers is greater than either. (c) The product of two rational numbers is not zero.

See Solution

Problem 328

List all factors of 60.

See Solution

Problem 329

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

See Solution

Problem 330

Find the prime factors of 90.

See Solution

Problem 331

Do even numbers have more factors than odd numbers? Explain your agreement with Milo or Lisa.

See Solution

Problem 332

Find the prime factorization of 300, using exponents for repeated factors: $300$ . What is the result?

See Solution

Problem 333

Judy's brother Sam has 96 comic books. What are the $t_{e_{n}}$ - ways he can divide them into equal groups?

See Solution

Problem 334

Find the prime factorization of 120 using a factor tree.

See Solution

Problem 335

Find the GCF of 72 and 36.

See Solution

Problem 336

Écris deux nombres premiers dont le produit est 589.

See Solution

Problem 337

Find the prime factorization of 3,600. Options: A. $2^{3} \times 3^{2} \times 5^{2}$ B. $2^{4} \times 3^{2} \times 5^{2}$ C. $2 \times 3 \times 5$ D. $2^{3} \times 3^{4} \times 5^{2}$

See Solution

Problem 338

Rewrite these integers as products of factors, including at least one perfect square:
1. $40=$
2. $75=$

See Solution

Problem 339

Find the least common multiple (LCM) of 18 and 42 using prime factorization. The LCM of 18 and 42 is $\text{LCM}(18, 42)$ .

See Solution

Problem 340

Find the GCF of 24 and 56 using prime factorization. The GCF is $$.

See Solution

Problem 341

Can a number not divisible by 3 be divisible by 6? Explain. Can a number not divisible by 6 be divisible by 3? Explain.

See Solution

Problem 342

Find the prime factorization for these: a. $1 \cdot 2 \cdot ... \cdot 12$ b. $35^{5} \cdot 65 \cdot 121^{15}$ c. 193 d. $400^{12}$

See Solution

Problem 343

If 14 divides $n$ , what other natural numbers divide $n$ ? Explain why.

See Solution

Problem 344

Find the prime decomposition of 1197 using index form from the prime factor tree provided.

See Solution

Problem 345

What does raising a number to the power of 0 mean? Find the value of $3^{0}$ based on the pattern in exponents.

See Solution

Problem 346

Find the prime factors of 16 and 18. Then calculate their LCM and GCF.
Prime factors: 16 = $18 =$
LCM = $GCF =$

See Solution

Problem 347

Which number is composite: A. 17, B. 23, C. 39, D. 41?

See Solution

Problem 348

Which of the following numbers is composite? A. 17 B. 23 C. 39 D. 41

See Solution

Problem 349

Is 37 prime? Todd says it's odd, Julian says it has 2 factors. Who is right? A. Todd B. Julian C. Both D. Neither

See Solution

Problem 350

Find the HCF of 30 and 42 as a product of prime factors. Then calculate the HCF.

See Solution

Problem 351

The student wrote $4 \times 5$ for the prime factorization of endangered amphibian species. Identify the mistake and provide the correct factorization.

See Solution

Problem 352

Find the LCM of 105 and 325 using prime factor trees.

See Solution

Problem 353

Prove by induction that $9^{2n} - 5^{2n}$ is divisible by 7 for positive integers $n$ .

See Solution

Problem 354

Find the highest common factor (HCF) of 66 and 110 using prime factor trees.

See Solution

Problem 355

Identify all prime numbers in the range from 10 to 20.

See Solution

Problem 356

Find the three prime numbers that multiply to give 231.

See Solution

Problem 357

Complete the prime factor trees for 21 and 33. What is the HCF of 21 and 33?

See Solution

Problem 358

Find the highest common factor (HCF) of 70 and 385 using their prime factors: 70 = $2 \times 5 \times 7$ , 385 = $5 \times 7 \times 11$ .

See Solution

Problem 359

Complete the prime factor trees for 21 and 33. Find the highest common factor (HCF) of 21 and 33.

See Solution

Problem 360

Tick each statement as always true, sometimes true, or never true:
1. Prime numbers are odd.
2. Prime numbers can have 3 or more factors.
3. The sum of 2 prime numbers is always even.
4. An array of a prime number will be incomplete.

See Solution

Problem 361

What is the power of 7 in the prime factorization of 98 expressed in index notation?

See Solution

Problem 362

Write 375 as a product of prime factors in index form.

See Solution

Problem 363

Find the prime factorization of 375 in index form and calculate the HCF of 375 and 150.

See Solution

Problem 364

Find the factors of 17.

See Solution

Problem 365

Identify the number that is not a perfect square: $\sqrt{9}$ , $\sqrt{200}$ , $\sqrt{4}$ .

See Solution

Problem 366

Which of these numbers is a square number? A) $4 \times 10^{5}$ B) $9 \times 10^{4}$ C) $4 \times 10^{3}$ D) $9 \times 10^{3}$

See Solution

Problem 367

3355432 equals $2^{25}$ . Why is it not a square number? (1 mark) Consider its prime factorization.

See Solution

Problem 368

Find the prime factorization of 700 in index form.

See Solution

Problem 369

Identify the non-prime factors of 42 from the list: 1, 2, 3, 6, 7, 14, 21, 42.

See Solution

Problem 370

Find the prime number from this list using divisibility tests: 69, 45, 38, 47, 52.

See Solution

Problem 371

Calculate the LCM of 16 and 18 using their prime factorizations: $16 = 2^4$ , $18 = 2^1 \cdot 3^2$ .

See Solution

Problem 372

Find the prime factors, LCM, and GCF of 42 and 68. List factors from least to greatest.

See Solution

Problem 373

Find the prime factorization of 40. $40 =$ $2^{3} \cdot 5$

See Solution

Problem 374

Find the prime factorization of 54. Is it $2 \cdot 3^{3}$ , $2^{2} \cdot 3^{3}$ , or $2 \cdot 2^{3} \cdot 3^{2}$ ?

See Solution

Problem 375

Find the LCM of 66 and 242 by determining their prime factors.
Prime factors of 66 = ? Prime factors of 242 = ? LCM of 66 and 242 = ?

See Solution

Problem 376

Find the prime factorization of 16. Options: $4^{3}$ , $2(2^{4})$ , $2^{5}$ , $2^{4}$ .

See Solution

Problem 377

Find the HCF and LCM of 315 ( $3^2 \cdot 5 \cdot 7$ ) and 693 ( $3^2 \cdot 7 \cdot 11$ ).

See Solution

Problem 378

Create prime factor trees for 40 and 220, then determine the LCM of 40 and 220 using those trees.

See Solution

Problem 379

Find the prime factor tree for 170 and use it to calculate the LCM of 105 and 170.

See Solution

Problem 380

Prove that the sum of two even numbers, $a$ and $b$ , is even: $a + b$ is even if $a = 2m$ and $b = 2n$ .

See Solution

Problem 381

Find $m$ and $n$ such that $96 = 2^{m} \times n$ , where $m$ and $n$ are prime numbers.

See Solution

Problem 382

The HCF of 30 and 75 is $\sqrt{x}$ . Find the value of $x$ .

See Solution

Problem 383

Find a whole number based on these clues:
1. $8 \leq 2x \leq 18$
2. $x$ is prime
3. $x$ is not a factor of 100.
What is $x$ ?

See Solution

Problem 384

Find two prime numbers that multiply to 85 and calculate their sum.

See Solution

Problem 385

Find the highest common factor (HCF) of 4725 and 5850, given their prime factorizations: $4725=3^{3} \times 5^{2} \times 7$ and $5850=2 \times 3^{2} \times 5^{2} \times 13$ .

See Solution

Problem 386

Complete the prime factor trees for 21 and 33. What is the highest common factor (HCF) of 21 and 33?

See Solution

Problem 387

Find the LCM of 1400 ( $2^3 \times 5^2 \times 7^1$ ) and 2100 ( $2^2 \times 3^1 \times 5^2 \times 7^1$ ) in prime factor form.

See Solution

Problem 388

Find three consecutive prime numbers that add up to 173. What is the largest of these numbers?

See Solution

Problem 389

Select two number sets with varying densities and describe why one set has a higher density than the other.

See Solution

Problem 390

Identify the irrational numbers from this list: -4.8237, $\frac{\pi}{2}$ , $\sqrt[3]{4}$ , $4+\sqrt{25}$ .

See Solution

Problem 391

See Solution

Problem 392

Eliza claims 30 is the only multiple of 3 that is also a multiple of 10. Is she right? Explain your reasoning.

See Solution

Problem 393

Find the prime decomposition of 1683 in index form using the provided prime factor tree.

See Solution

Problem 394

Draw the prime factor tree for 60 and find the HCF of 60 and 468 using it.

See Solution

Problem 395

Draw prime factor trees for 63 and 105, then use them to find the highest common factor (HCF) of 63 and 105.

See Solution

Problem 396

Find the HCF of 70 and 385 using prime factor trees. Given: $385 \rightarrow 5 \times 77$ . Factor 70 to find HCF.

See Solution

Problem 397

Find the lowest common multiple (LCM) of 130 and 165 using their prime factor trees.

See Solution

Problem 398

Evaluate $8^{4}$ . Choose if $\pi$ is rational or irrational.

See Solution

Problem 399

Draw prime factor trees for 42 and 66, then find their lowest common multiple (LCM).

See Solution

Problem 400

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

See Solution

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Number Theory

Problem 301

Show that if p,q∈Np, q \in \mathbb{N}p,q∈N and pq=1−12+13−⋯+11319\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\cdots+\frac{1}{1319}qp​=1−21​+31​−⋯+13191​, then ppp is divisible by 1979.

Problem 302

Draw the prime factor tree for 190 and find the LCM of 105 and 190.

Problem 303

Problem 304

Find the prime factorization of 396 in index form and identify which of the following numbers are factors: 88, 14, 9, 121, 22.

Problem 305

Find the prime factorization of 192 and list the factors in ascending order.

Problem 306

Express 80 as a product of prime factors in ascending order.

Problem 307

Find five pairs of prime numbers that add up to 48.

Problem 308

Identify which of the following numbers are factors of 28,728, given its prime factorization: 23×33×7×192^{3} \times 3^{3} \times 7 \times 1923×33×7×19.1. 63=32×763=3^{2} \times 763=32×72. 16=2416=2^{4}16=24

Problem 309

The LCM of two numbers has two 222s and three 333s. What can you deduce about these numbers?

Problem 310

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

Problem 311

Find the minimum 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 312

Amiyah landed on a prime number between 1 and 50. Which number could NOT be it: 43, 31, 21, or 13?

Problem 313

Find the least common multiple of A=3⋅5⋅17⋅37A=3 \cdot 5 \cdot 17 \cdot 37A=3⋅5⋅17⋅37 and B=33⋅5⋅113⋅37B=3^3 \cdot 5 \cdot 11^3 \cdot 37B=33⋅5⋅113⋅37.

Problem 314

How many digits does the product of 282589933−12^{82589933}-1282589933−1 and 11 have beyond 24862048 digits?

Problem 315

Find all numbers that are multiples of 30.

Problem 316

List all the factors of 818181.

Problem 317

Circle the composite numbers from this list: 15, 2, 16, 17, 29, 27, 8, 10, 0, 9, 6, 11, 3, 20. Composite numbers have more than 2 factors.

Problem 318

Find the factors of m=25×x×y3m = 2^{5} \times x \times y^{3}m=25×x×y3, where xxx and yyy are primes > 2. Select all that apply: x2,8,xy,2y4,4y2,64xx^{2}, 8, xy, 2y^{4}, 4y^{2}, 64xx2,8,xy,2y4,4y2,64x.

Problem 319

Find the prime factorization of 240. Also, determine if the factorization of 90 is prime.

Problem 320

Find the prime factorization of 240. Use the numbers: 2∧42^{\wedge} 42∧4, 3∧13^{\wedge} 13∧1, 5∧15^{\wedge} 15∧1.

Problem 321

Is the prime factorization of 90 correct? Choose from: 5⋅2⋅95 \cdot 2 \cdot 95⋅2⋅9, 2⋅3⋅152 \cdot 3 \cdot 152⋅3⋅15, or 2⋅3⋅3⋅52 \cdot 3 \cdot 3 \cdot 52⋅3⋅3⋅5.

Problem 322

Identify the relatively prime pair from these options: A) 72 and 115 B) 120 and 835 C) 201 and 735 D) 303 and 1041.

Problem 323

Find the greatest length in meters for equal pieces from 63 m of pink and 105 m of green ribbon with no leftovers.

Problem 324

Find the GCD of 30 and 42, and show it as a product of its prime factors: 30=2⋅3⋅530 = 2 \cdot 3 \cdot 530=2⋅3⋅5, 42=2⋅3⋅742 = 2 \cdot 3 \cdot 742=2⋅3⋅7.

Problem 325

a) Count the prime numbers that are multiples of 5. b) Count the prime numbers that are multiples of 8.

Problem 326

Find the common factors of 56 and 28. What is the value of 561×56−x+281×28−x\frac{\frac{56}{1 \times 56}}{-x}+\frac{\frac{28}{1 \times 28}}{-x}−x1×5656​​+−x1×2828​​?

Problem 327

Determine if these statements are always, sometimes, or never true: (a) The square of a non-zero rational number is negative. (b) The product of two negative rational numbers is greater than either. (c) The product of two rational numbers is not zero.

Problem 328

List all factors of 60.

Problem 329

Identify the correct prime factorization of 36 in exponential form: 4⋅94 \cdot 94⋅9, 626^{2}62, 22⋅322^{2} \cdot 3^{2}22⋅32, 2⋅322 \cdot 3^{2}2⋅32.

Problem 330

Find the prime factors of 90.

Problem 331

Do even numbers have more factors than odd numbers? Explain your agreement with Milo or Lisa.

Problem 332

Find the prime factorization of 300, using exponents for repeated factors: 300300300. What is the result?

Problem 333

Judy's brother Sam has 96 comic books. What are the tent_{e_{n}}ten​​ - ways he can divide them into equal groups?

Problem 334

Find the prime factorization of 120 using a factor tree.

Problem 335

Find the GCF of 72 and 36.

Problem 336

Écris deux nombres premiers dont le produit est 589.

Problem 337

Find the prime factorization of 3,600. Options: A. 23×32×522^{3} \times 3^{2} \times 5^{2}23×32×52 B. 24×32×522^{4} \times 3^{2} \times 5^{2}24×32×52 C. 2×3×52 \times 3 \times 52×3×5 D. 23×34×522^{3} \times 3^{4} \times 5^{2}23×34×52

Problem 338

Rewrite these integers as products of factors, including at least one perfect square: 1. 40=40=40= 2. 75=75=75=

Problem 339

Find the least common multiple (LCM) of 18 and 42 using prime factorization. The LCM of 18 and 42 is LCM(18,42)\text{LCM}(18, 42)LCM(18,42).

Problem 340

Find the GCF of 24 and 56 using prime factorization. The GCF is $$.

Show that if $p, q \in \mathbb{N}$ and $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\cdots+\frac{1}{1319}$ , then $p$ is divisible by 1979.

Identify which of the following numbers are factors of 28,728, given its prime factorization: $2^{3} \times 3^{3} \times 7 \times 19$ .
1. $63=3^{2} \times 7$
2. $16=2^{4}$

The LCM of two numbers has two $2$ s and three $3$ s. What can you deduce about these numbers?

Find the minimum 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.

Find the least common multiple of $A=3 \cdot 5 \cdot 17 \cdot 37$ and $B=3^3 \cdot 5 \cdot 11^3 \cdot 37$ .

How many digits does the product of $2^{82589933}-1$ and 11 have beyond 24862048 digits?

List all the factors of $81$ .

Find the factors of $m = 2^{5} \times x \times y^{3}$ , where $x$ and $y$ are primes > 2. Select all that apply: $x^{2}, 8, xy, 2y^{4}, 4y^{2}, 64x$ .

Find the prime factorization of 240. Use the numbers: $2^{\wedge} 4$ , $3^{\wedge} 1$ , $5^{\wedge} 1$ .

Is the prime factorization of 90 correct? Choose from: $5 \cdot 2 \cdot 9$ , $2 \cdot 3 \cdot 15$ , or $2 \cdot 3 \cdot 3 \cdot 5$ .

Find the GCD of 30 and 42, and show it as a product of its prime factors: $30 = 2 \cdot 3 \cdot 5$ , $42 = 2 \cdot 3 \cdot 7$ .

Find the common factors of 56 and 28. What is the value of $\frac{\frac{56}{1 \times 56}}{-x}+\frac{\frac{28}{1 \times 28}}{-x}$ ?

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

Find the prime factorization of 300, using exponents for repeated factors: $300$ . What is the result?

Judy's brother Sam has 96 comic books. What are the $t_{e_{n}}$ - ways he can divide them into equal groups?

Find the prime factorization of 3,600. Options: A. $2^{3} \times 3^{2} \times 5^{2}$ B. $2^{4} \times 3^{2} \times 5^{2}$ C. $2 \times 3 \times 5$ D. $2^{3} \times 3^{4} \times 5^{2}$

Rewrite these integers as products of factors, including at least one perfect square:
1. $40=$
2. $75=$

Find the least common multiple (LCM) of 18 and 42 using prime factorization. The LCM of 18 and 42 is $\text{LCM}(18, 42)$ .

Find the prime factorization for these: a. $1 \cdot 2 \cdot ... \cdot 12$ b. $35^{5} \cdot 65 \cdot 121^{15}$ c. 193 d. $400^{12}$

If 14 divides $n$ , what other natural numbers divide $n$ ? Explain why.

What does raising a number to the power of 0 mean? Find the value of $3^{0}$ based on the pattern in exponents.

Find the prime factors of 16 and 18. Then calculate their LCM and GCF.
Prime factors: 16 = $18 =$
LCM = $GCF =$

The student wrote $4 \times 5$ for the prime factorization of endangered amphibian species. Identify the mistake and provide the correct factorization.

Prove by induction that $9^{2n} - 5^{2n}$ is divisible by 7 for positive integers $n$ .

Find the highest common factor (HCF) of 70 and 385 using their prime factors: 70 = $2 \times 5 \times 7$ , 385 = $5 \times 7 \times 11$ .

Tick each statement as always true, sometimes true, or never true:
1. Prime numbers are odd.
2. Prime numbers can have 3 or more factors.
3. The sum of 2 prime numbers is always even.
4. An array of a prime number will be incomplete.

Identify the number that is not a perfect square: $\sqrt{9}$ , $\sqrt{200}$ , $\sqrt{4}$ .

Which of these numbers is a square number? A) $4 \times 10^{5}$ B) $9 \times 10^{4}$ C) $4 \times 10^{3}$ D) $9 \times 10^{3}$

3355432 equals $2^{25}$ . Why is it not a square number? (1 mark) Consider its prime factorization.

Calculate the LCM of 16 and 18 using their prime factorizations: $16 = 2^4$ , $18 = 2^1 \cdot 3^2$ .

Find the prime factorization of 40. $40 =$ $2^{3} \cdot 5$

Find the prime factorization of 54. Is it $2 \cdot 3^{3}$ , $2^{2} \cdot 3^{3}$ , or $2 \cdot 2^{3} \cdot 3^{2}$ ?