Number Theory

Problem 501

Find the highest common factor (HCF) of 315 and 4125, given their prime factorizations: $315=3^{2} \times 5 \times 7$ and $4125=3 \times 5^{3} \times 11$ .

See Solution

Problem 502

Find the LCM of the numbers with prime factorizations: $84=2^{2} \times 3 \times 7$ and $270=2 \times 3^{3} \times 5$ .

See Solution

Problem 503

Create prime factor trees for 42 and 66, then use them to calculate the LCM of 42 and 66.

See Solution

Problem 504

Find the lowest common multiple (LCM) of 60 ( $2^{2} \times 3 \times 5$ ) and 378 ( $2 \times 3^{3} \times 7$ ).

See Solution

Problem 505

Create prime factor trees for 56 and 308, then use them to calculate the LCM of 56 and 308.

See Solution

Problem 506

Find the HCF of $5 \cdot m$ and $3 \cdot n$ where $m=3^{4} \times 5^{3}$ and $n=3^{3} \times 5^{2} \times 11$ .

See Solution

Problem 507

Find the greatest length in metres (m) for equal pieces from 44 m blue and 110 m red ribbon with no leftovers.

See Solution

Problem 508

Find the least common multiple (PPCM) and greatest common divisor (PGCD) for the following pairs and sets of numbers.

See Solution

Problem 509

Find the prime factorization of 455 using exponents for repeated factors, like $5^{\wedge} 7^{\star}$ .

See Solution

Problem 510

Find the prime factorization of 84 using exponents for repeated factors, like $2^{\wedge} 2^{\star} 3$ .

See Solution

Problem 511

Determine the missing prime factors in the factor tree for 105.

See Solution

Problem 512

Find the prime number between 95 and 100. Options: a. 96, b. 97, c. 98, d. 99.

See Solution

Problem 513

Miriam's uncle donates 120 juice cans and $90 \mathrm{p}$ packs of crackers.
a. Find the max students sharing equally and their share.
b. If he eats 2 packs, find the new max students and their share.

See Solution

Problem 514

Describe the numbers 25, 31, 51, and 1 as prime, composite, or square.

See Solution

Problem 515

Complete the prime factor tree for 130 and write its prime factorization: $130 = 2 \times 5 \times 13$ .

See Solution

Problem 516

Find all possible group sizes for 45 students on a zoo trip, ensuring no group has only one student.

See Solution

Problem 517

What is the smallest number with exactly four different prime factors?

See Solution

Problem 518

Find the smallest positive integer $h$ such that both 60 and 63 are factors of $h$ .

See Solution

Problem 519

Find which of the following numbers are factors of 16,632, given its prime factorization $2^{3} \times 3^{3} \times 7 \times 11$ :
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 520

Find the greatest number of vases Corey can use to evenly distribute 20 poppies and 28 marigolds.

See Solution

Problem 521

Identify common primes in the factor trees of 30 and 42, find the HCF as prime factors, and calculate the HCF.

See Solution

Problem 522

How many squares in each row for these items: a. Waffle (16), b. Magic square (49), c. Tile game (100)? Find factors.

See Solution

Problem 523

Find the different ways to express the number 30 mathematically.

See Solution

Problem 524

Find the HCF of 825 and 950 using prime factorization: $825 = 3^a \times 5^b \times 11^c$ , $950 = 2^d \times 5^e \times 19^f$ .

See Solution

Problem 525

Find the LCM of 204 and 340 using their prime factors.

See Solution

Problem 526

Draw the prime factor tree for 90 and find the HCF of 90 and 396 using it.

See Solution

Problem 527

Find the missing numbers in the prime factor tree of 14. What are the prime factors of 14?

See Solution

Problem 528

Determine $a$ and $b$ for the prime factorization of 324 as $2^{a} \cdot 3^{b}$ .

See Solution

Problem 529

Show that for any natural number $n \neq 1$ , we have $7 \mid (n^{7} - n)$ .

See Solution

Problem 530

Find the prime decomposition of 1881. Is it divisible by 3, 11, and 19? Write the decomposition in index form: $1881 = 3^2 \times 11^1 \times 19^1$ .

See Solution

Problem 531

Prove that there is no rational number $r$ for which $r^{2}=3$ .

See Solution

Problem 532

Verify if all numbers divisible by 100 are also divisible by 5 using deductive reasoning or a counterexample.

See Solution

Problem 533

Find the GCF of 42 and 70. The answer is $$.

See Solution

Problem 534

Show that for any natural number $n \neq 1$ , $7$ divides $n^{7} - n$ .

See Solution

Problem 535

Identify the prime number from this list: 18, 6, 10, 11, 9.

See Solution

Problem 536

A spinner has 30 sections numbered 1 to 30.
A: Find the probability of landing on a factor of 30 as a fraction.
B: Find the same probability as a percent, rounded to the nearest tenth.

See Solution

Problem 537

How many prime numbers are also multiples of 5?

See Solution

Problem 538

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. $16=2^{4}$
3. $8=2^{3}$
4. $459=3^{3} \times 17$
5. $38=2 \times 19$

See Solution

Problem 539

Find which of the following numbers are factors of 16,632, given its prime factorization: $2^{3} \times 3^{3} \times 7 \times 11$ .
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 540

How many prime numbers are multiples of 5 and multiples of 8?

See Solution

Problem 541

Find the highest common factor (HCF) of 42 and 231 using their prime factor trees.

See Solution

Problem 542

If $x$ gives a remainder of 4 when divided by 7, what is the remainder when $x$ is divided by 4?

See Solution

Problem 543

Find the greatest length in metres (m) for equal pieces from 99 m yellow and 165 m purple ribbon.

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Problem 544

Identify the common prime factors of 6435 and 6930 from the options: $3^{2}$ , $3^{2} \times 5 \times 11$ , $2 \times 3$ , $3 \times 11$ , $2 \times 3^{4} \times 5^{2} \times 7 \times 11^{2} \times 13$ , $2 \times 3 \times 5 \times 7 \times 11 \times 13$ , $2 \times 7 \times 13$ .

See Solution

Problem 545

Find the HCF and LCM of 60 (factors: 2, 2, 3, 5) and 220 (factors: 2, 2, 5, 11).

See Solution

Problem 546

Find the greatest length in meters for equal pieces that Damian can cut from 63 m of pink ribbon and 105 m of green ribbon.

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Problem 547

Geoff has blood tests every 70 days and fitness tests every 75 days. When is the next day he has both tests?

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Problem 548

Find the smallest positive integer $m$ that has both 28 and 90 as factors.

See Solution

Problem 549

Find the prime factors of 84.

See Solution

Problem 550

Find the prime factorization of 88. If it's prime, state that.

See Solution

Problem 551

Find the highest common factor (HCF) of 42 and 231 using their prime factor trees.

See Solution

Problem 552

Create prime factor trees for 70 and 245, then determine the highest common factor (HCF) of 70 and 245.

See Solution

Problem 553

Identify a counterexample to the conjecture: "All prime numbers are odd." Options: A. 1 B. 2 C. 3 D. 4

See Solution

Problem 554

Explain why $m n$ is not always the least common multiple of $m$ and $n$ .

See Solution

Problem 555

Find the prime factorization of 196, using exponents for repeated factors.

See Solution

Problem 556

Сколько трехзначных номеров без цифры 8 можно составить для велосипедного клуба?

See Solution

Problem 557

Find the factors of 4 and determine if 4 is a prime number based on your findings.

See Solution

Problem 558

Find the greatest common factor to simplify $\frac{36}{90}$ . A. 2 B. 6 C. 9 D. 18

See Solution

Problem 559

Soit $P$ un nombre premier. Est-ce que $P^{2}$ est premier?

See Solution

Problem 560

Li rolls two 4-sided dice and multiplies the results. What is the probability that the score is a prime number?

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Problem 561

Find the prime factors of 72.

See Solution

Problem 562

Find the prime factorization in exponential form of 48.

See Solution

Problem 563

Find the prime factorization of 100, knowing one factor is 2. Express it as a product of primes: $100 = 2 \times 2 \times \ldots$

See Solution

Problem 564

Find the prime factors of 18 and 24, and express each as a product of primes.

See Solution

Problem 565

Express 900 as a product of its prime factors in index form: find the prime factors of 900 in ascending order.

See Solution

Problem 566

Express 900 as a product of its prime factors in index form, listing them in ascending order.

See Solution

Problem 567

Find the Highest Common Factor (HCF) of 600 and 1050.

See Solution

Problem 568

Find all factors of 12 and determine the highest common factor (HCF) of 18 and 12.

See Solution

Problem 569

Find the highest common factor (HCF) of 22 and 55.

See Solution

Problem 570

List the factors of 10 and 14. What is the highest common factor (HCF) of 10 and 14?

See Solution

Problem 571

Find the common prime factors of 6930 and 9009 from the options below:
1. $3^{2} \times 7 \times 11$
2. $2 \times 3$
3. $2 \times 5 \times 13$
4. $2 \times 3^{4} \times 5 \times 7^{2} \times 11^{2} \times 13$
5. $2 \times 3 \times 5 \times 7 \times 11 \times 13$
6. $3^{2}$
7. $3 \times 11$

See Solution

Problem 572

Find the largest number that is a divisor of both 36 and 48.

See Solution

Problem 573

Find the smallest number of students that can be grouped into 9 and 12, but leaves a remainder when grouped into 11. How many are left over?

See Solution

Problem 574

Find the LCM of 7 and 5.

See Solution

Problem 575

Find the prime factorization of 1287 using the tree: $1287 = 3^2 \times 11^1 \times 13^1$ .

See Solution

Problem 576

Ring the numbers that are divisible by 9: $1, \quad 9, \quad 27, \quad 56, \quad 99$ .

See Solution

Problem 577

How many factors does 72 have? Calculate the factors of 72.

See Solution

Problem 578

Identify the following from the numbers 2, 9, 14, 36, 40: a) a multiple of 12, b) a prime number, c) a factor of 64.

See Solution

Problem 579

Find two prime numbers that multiply to 21 and calculate their sum.

See Solution

Problem 580

Find two numbers greater than 8 with HCF of 8 and LCM of 80.

See Solution

Problem 581

Find two prime numbers that sum to 30. What is the difference between these two primes?

See Solution

Problem 582

The HCF of two numbers is 21. Identify the two prime factors present in both numbers' decompositions.

See Solution

Problem 583

Find the common prime factors of 6435 and 6930 from the given decompositions: $6435=3^{2} \times 5 \times 11 \times 13$ and $6930=2 \times 3^{2} \times 5 \times 7 \times 11$ .

See Solution

Problem 584

Is $2^{2} \times 7^{1}$ a factor of $a=2^{3} \times 3^{1} \times 7^{2}$ ? Explain your reasoning.

See Solution

Problem 585

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

See Solution

Problem 586

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

See Solution

Problem 587

Find the HCF of 70 and 385 using their prime factor trees. For 70: $2 \times 5 \times 7$ and for 385: $5 \times 7 \times 11$ .

See Solution

Problem 588

Complete the prime factor trees for 42 and 105. Find the highest common factor (HCF) of 42 and 105.

See Solution

Problem 589

Identify the prime numbers in the range from 50 to 60.

See Solution

Problem 590

Find the LCM of 70 and 273 using their prime factor trees: 70 = $2 \times 5 \times 7$ , 273 = $3 \times 7 \times 13$ .

See Solution

Problem 591

Find the highest common factor (HCF) of 330 and 385 using their prime factor trees.

See Solution

Problem 592

Which numbers below are factors of 19,656 given its prime factorization $2^{3} \times 3^{3} \times 7 \times 13$ ?
1. $8=2^{3}$
2. $297=3^{3} \times 11$
3. $63=3^{2} \times 7$
4. $16=2^{4}$
5. $14=2 \times 7$

See Solution

Problem 593

Determine divisibility for the numbers 162, 245, 430, 609, 684:
1. Divisible by 2?
2. Divisible by 3?
3. Divisible by 5?
4. Is an even number divisible by 3 also divisible by 6? Explain.

See Solution

Problem 594

Find the LCM of 105 (factors: 3, 5, 7) and 170 (factors: 2, 5, 17).

See Solution

Problem 595

Find the LCM of 154 and 273 using their corrected prime factors: 154 = 2 \times 7 \times 11 and 273 = 3 \times 7 \times 13.

See Solution

Problem 596

Find the LCM of 42 and 66 using prime factor trees.

See Solution

Problem 597

Is 3,213 divisible by 9? Verify the conjecture using the fact that digits summing to 9 indicates divisibility.

See Solution

Problem 598

Find the factors of 9 and determine if 9 is a prime number.

See Solution

Problem 599

Determine if each number is prime or not: a) 7, b) 1, c) 8.

See Solution

Problem 600

Find all the factor pairs of 12: $12=\square \times \square 12=\square \times \square 12=\square \times \square$

See Solution

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Number Theory

Problem 501

Find the highest common factor (HCF) of 315 and 4125, given their prime factorizations: 315=32×5×7315=3^{2} \times 5 \times 7315=32×5×7 and 4125=3×53×114125=3 \times 5^{3} \times 114125=3×53×11.

Problem 502

Find the LCM of the numbers with prime factorizations: 84=22×3×784=2^{2} \times 3 \times 784=22×3×7 and 270=2×33×5270=2 \times 3^{3} \times 5270=2×33×5.

Problem 503

Create prime factor trees for 42 and 66, then use them to calculate the LCM of 42 and 66.

Problem 504

Find the lowest common multiple (LCM) of 60 (22×3×52^{2} \times 3 \times 522×3×5) and 378 (2×33×72 \times 3^{3} \times 72×33×7).

Problem 505

Create prime factor trees for 56 and 308, then use them to calculate the LCM of 56 and 308.

Problem 506

Find the HCF of 5⋅m5 \cdot m5⋅m and 3⋅n3 \cdot n3⋅n where m=34×53m=3^{4} \times 5^{3}m=34×53 and n=33×52×11n=3^{3} \times 5^{2} \times 11n=33×52×11.

Problem 507

Find the greatest length in metres (m) for equal pieces from 44 m blue and 110 m red ribbon with no leftovers.

Problem 508

Find the least common multiple (PPCM) and greatest common divisor (PGCD) for the following pairs and sets of numbers.

Problem 509

Find the prime factorization of 455 using exponents for repeated factors, like 5∧7⋆5^{\wedge} 7^{\star}5∧7⋆.

Problem 510

Find the prime factorization of 84 using exponents for repeated factors, like 2∧2⋆32^{\wedge} 2^{\star} 32∧2⋆3.

Problem 511

Determine the missing prime factors in the factor tree for 105.

Problem 512

Find the prime number between 95 and 100. Options: a. 96, b. 97, c. 98, d. 99.

Problem 513

Miriam's uncle donates 120 juice cans and 90p90 \mathrm{p}90p packs of crackers. a. Find the max students sharing equally and their share.b. If he eats 2 packs, find the new max students and their share.

Problem 514

Describe the numbers 25, 31, 51, and 1 as prime, composite, or square.

Problem 515

Complete the prime factor tree for 130 and write its prime factorization: 130=2×5×13130 = 2 \times 5 \times 13130=2×5×13.

Problem 516

Find all possible group sizes for 45 students on a zoo trip, ensuring no group has only one student.

Problem 517

What is the smallest number with exactly four different prime factors?

Problem 518

Find the smallest positive integer hhh such that both 60 and 63 are factors of hhh.

Problem 519

Problem 520

Find the greatest number of vases Corey can use to evenly distribute 20 poppies and 28 marigolds.

Problem 521

Identify common primes in the factor trees of 30 and 42, find the HCF as prime factors, and calculate the HCF.

Problem 522

How many squares in each row for these items: a. Waffle (16), b. Magic square (49), c. Tile game (100)? Find factors.

Problem 523

Find the different ways to express the number 30 mathematically.

Problem 524

Find the HCF of 825 and 950 using prime factorization: 825=3a×5b×11c825 = 3^a \times 5^b \times 11^c825=3a×5b×11c, 950=2d×5e×19f950 = 2^d \times 5^e \times 19^f950=2d×5e×19f.

Problem 525

Find the LCM of 204 and 340 using their prime factors.

Problem 526

Draw the prime factor tree for 90 and find the HCF of 90 and 396 using it.

Problem 527

Find the missing numbers in the prime factor tree of 14. What are the prime factors of 14?

Problem 528

Determine aaa and bbb for the prime factorization of 324 as 2a⋅3b2^{a} \cdot 3^{b}2a⋅3b.

Problem 529

Show that for any natural number n≠1n \neq 1n=1, we have 7∣(n7−n)7 \mid (n^{7} - n)7∣(n7−n).

Problem 530

Find the prime decomposition of 1881. Is it divisible by 3, 11, and 19? Write the decomposition in index form: 1881=32×111×1911881 = 3^2 \times 11^1 \times 19^11881=32×111×191.

Problem 531

Prove that there is no rational number rrr for which r2=3r^{2}=3r2=3.

Problem 532

Verify if all numbers divisible by 100 are also divisible by 5 using deductive reasoning or a counterexample.

Problem 533

Find the GCF of 42 and 70. The answer is $$.

Problem 534

Show that for any natural number n≠1n \neq 1n=1, 777 divides n7−nn^{7} - nn7−n.

Problem 535

Identify the prime number from this list: 18, 6, 10, 11, 9.

Problem 536

A spinner has 30 sections numbered 1 to 30. A: Find the probability of landing on a factor of 30 as a fraction. B: Find the same probability as a percent, rounded to the nearest tenth.

Problem 537

How many prime numbers are also multiples of 5?

Problem 538

Problem 539

Problem 540

How many prime numbers are multiples of 5 and multiples of 8?

Problem 541

Find the highest common factor (HCF) of 42 and 231 using their prime factor trees.

Find the highest common factor (HCF) of 315 and 4125, given their prime factorizations: $315=3^{2} \times 5 \times 7$ and $4125=3 \times 5^{3} \times 11$ .

Find the LCM of the numbers with prime factorizations: $84=2^{2} \times 3 \times 7$ and $270=2 \times 3^{3} \times 5$ .

Find the lowest common multiple (LCM) of 60 ( $2^{2} \times 3 \times 5$ ) and 378 ( $2 \times 3^{3} \times 7$ ).

Find the HCF of $5 \cdot m$ and $3 \cdot n$ where $m=3^{4} \times 5^{3}$ and $n=3^{3} \times 5^{2} \times 11$ .

Find the prime factorization of 455 using exponents for repeated factors, like $5^{\wedge} 7^{\star}$ .

Find the prime factorization of 84 using exponents for repeated factors, like $2^{\wedge} 2^{\star} 3$ .

Miriam's uncle donates 120 juice cans and $90 \mathrm{p}$ packs of crackers.
a. Find the max students sharing equally and their share.
b. If he eats 2 packs, find the new max students and their share.

Complete the prime factor tree for 130 and write its prime factorization: $130 = 2 \times 5 \times 13$ .

Find the smallest positive integer $h$ such that both 60 and 63 are factors of $h$ .

Find the HCF of 825 and 950 using prime factorization: $825 = 3^a \times 5^b \times 11^c$ , $950 = 2^d \times 5^e \times 19^f$ .

Determine $a$ and $b$ for the prime factorization of 324 as $2^{a} \cdot 3^{b}$ .

Show that for any natural number $n \neq 1$ , we have $7 \mid (n^{7} - n)$ .

Find the prime decomposition of 1881. Is it divisible by 3, 11, and 19? Write the decomposition in index form: $1881 = 3^2 \times 11^1 \times 19^1$ .

Prove that there is no rational number $r$ for which $r^{2}=3$ .

Show that for any natural number $n \neq 1$ , $7$ divides $n^{7} - n$ .

A spinner has 30 sections numbered 1 to 30.
A: Find the probability of landing on a factor of 30 as a fraction.
B: Find the same probability as a percent, rounded to the nearest tenth.

If $x$ gives a remainder of 4 when divided by 7, what is the remainder when $x$ is divided by 4?

Find the smallest positive integer $m$ that has both 28 and 90 as factors.

Explain why $m n$ is not always the least common multiple of $m$ and $n$ .

Find the greatest common factor to simplify $\frac{36}{90}$ . A. 2 B. 6 C. 9 D. 18

Soit $P$ un nombre premier. Est-ce que $P^{2}$ est premier?

Find the prime factorization of 100, knowing one factor is 2. Express it as a product of primes: $100 = 2 \times 2 \times \ldots$