Number Theory

Problem 601

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

See Solution

Problem 602

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

See Solution

Problem 603

Find the LCM of 120 ( $2^{3} \times 3 \times 5$ ) and 198 ( $2 \times 3^{2} \times 11$ ).

See Solution

Problem 604

Find the unknown unit symbol for the number of oxygen atoms in a $\mathrm{NO}_{2}$ molecule: amount or number of 1 one a or $n$ .

See Solution

Problem 605

Which sets $N, Z, Q, R$ does $\sqrt{5}$ belong to? Options: Only $Q$ , Only $Q$ and $R$ , Only $R$ , Only $Z$ .

See Solution

Problem 606

Identify the true statement among these:
1. An integer not a natural number exists.
2. All real numbers are rational.
3. Some rational numbers aren't real.
4. All integers except 0 are natural.

See Solution

Problem 607

Find the prime factorization of the number 32.

See Solution

Problem 608

Find the prime decomposition of 224 in the form $2^{a} \times b$ and determine the values of $a$ and $b$ .

See Solution

Problem 609

Which statement is true? A. Even + 3 is even. B. Even + 3 is prime. C. Even + 3 is odd. D. Prime + 3 is even.

See Solution

Problem 610

Write 504 as a product of its prime factors in the form $p_1^{e_1} \times p_2^{e_2} \times \ldots$

See Solution

Problem 611

Find the prime factorization of 252 in index form and identify which two of the following are factors of 252:
1) $22=2 \times 11$ 2) $24=2^{3} \times 3$ 3) $9=3^{2}$ 4) $14=2 \times 7$ 5) $49=7^{2}$ .

See Solution

Problem 612

Which numbers are factors of 25,704, given its prime factorization $2^{3} \times 3^{3} \times 7 \times 17$ ? Check all that apply:
63: $3^{2} \times 7$ , 513: $3^{3} \times 19$ , 34: $2 \times 17$ , 8: $2^{3}$ , 16: $2^{4}$ .

See Solution

Problem 613

Find the HCF of 60 and 114.

See Solution

Problem 614

Find the GCD of 36 and 55 using the Euclidean algorithm, then express it as an integral combination and find the inverse of $36 \bmod 55$ .

See Solution

Problem 615

Find the smallest composite number greater than 1 that is not divisible by $2$ , $3$ , $5$ , or $7$ .

See Solution

Problem 616

Count how many prime numbers can be formed using the digits 1 to $n$ (where $n$ is from 1 to 9) exactly once.

See Solution

Problem 617

Find a positive integer with 8 divisors, where two divisors are 21 and 33.

See Solution

Problem 618

Determine if the following numbers are prime or composite: 44, 12, 50, 83, 27, 53, 24, 67, 78.

See Solution

Problem 619

Determine if the following numbers are prime or composite: 44, 12, 50, 83, 53, 24, 34, 27, 67, 78.

See Solution

Problem 620

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

See Solution

Problem 621

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

See Solution

Problem 622

Find the common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 2240 ( $2^{6} \times 5 \times 7$ ) from the options.

See Solution

Problem 623

Find the prime factors of 52. The prime factorization is $2^2 \times 13$ .

See Solution

Problem 624

Find the prime factors of 102. The prime factorization is $2 \times 3 \times 17$ .

See Solution

Problem 625

Express 52 as a product of its prime factors. $52 =$

See Solution

Problem 626

Find the prime factorization of 64.

See Solution

Problem 627

Find the greatest length in meters for equal pieces from 44 m blue ribbon and 110 m red ribbon with no leftover.

See Solution

Problem 628

Find the GCF of 62 and 70 using prime factorization. Show your work for full credit.

See Solution

Problem 629

Find the prime factorization of 145.

See Solution

Problem 630

Czy $2^{34}+3^{56}$ jest liczbą pierwszą? Uzasadnij swoją odpowiedź.

See Solution

Problem 631

Find the number of factors of 40.

See Solution

Problem 632

Identify the relatively prime pair: A) 5 and 75 B) 12 and 51 C) 7 and 20 D) 15 and 21

See Solution

Problem 633

Identify the pair of numbers that are relatively prime: A) 119 & 209, B) 88 & 121, C) 144 & 165, D) 42 & 385.

See Solution

Problem 634

Find the common primes in the factor trees of 30 and 42, then determine their HCF as a product of primes and its value.

See Solution

Problem 635

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

See Solution

Problem 636

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

See Solution

Problem 637

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

See Solution

Problem 638

Identify the common prime factors of 6435 and 8190 from the options below: 2 × 3 × 5 × 7 × 11 × 13, 3 × 13, 2 × 3, 2 × 3^{4} × 5^{2} × 7 × 11 × 13^{2}, 3^{2} × 5 × 13, 2 × 7 × 11, 3^{2}.

See Solution

Problem 639

Identify the common prime factors of 6930 and 9009 from the options:
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 640

Find the prime factors of 105 using a tree and then calculate the HCF of 30 and 105.

See Solution

Problem 641

Identify the correct prime factorization of 200 in exponential form: $2^{3} \cdot 5^{2}$ , $2 \cdot 10^{2}$ , $2^{2} \cdot 5^{2}$ , or $8 \cdot 25$ .

See Solution

Problem 642

Décompose 124 en un produit de facteurs premiers.

See Solution

Problem 643

Find the prime factors of 36 in ascending order. Remember, 1 is not prime. What number goes in the green box?

See Solution

Problem 644

8是一个完全平方数吗？是否重看提交

See Solution

Problem 645

Find the prime factorization of 14 and list the factors in ascending order, like $2 \times 7$ .

See Solution

Problem 646

Why is there a 3 outside the cube root in $\sqrt[3]{54} = 3 \sqrt[3]{2}$ ? Explain its significance.

See Solution

Problem 647

Determine the prime factors of $50,225,675$ .

See Solution

Problem 648

How many numbers are in the intersection of Set A (prime squares) and Set B (multiples of 4)? A. 0 B. 1 C. 2 D. 3 E. 4

See Solution

Problem 649

Which of these numbers is NOT a multiple of $17$ : 17, 68, 117, 1700, or 3434?

See Solution

Problem 650

Find the smallest prime number from the sum of three different primes.

See Solution

Problem 651

Do 11 and 15 share exactly one common factor? Circle Yes or No and explain your reasoning.

See Solution

Problem 652

Optimize code that checks if $num1$ is divisible by $num2$ or both are zero. Discuss division, modulo, and conditionals.

See Solution

Problem 653

20. (a) Define what it means for an integer $a$ to divide another integer $b$ . (b) Prove that $7$ divides $12^{n}-5^{n}$ for all $n \in \mathbf{N}$ .

See Solution

Problem 654

Which statement is true: a) All integers are signed, b) All fractions are rational, c) All decimals are rational, d) Whole numbers are natural? Also, which is not a natural number: a) 3, b) -2, c) 5, d) 1000?

See Solution

Problem 655

Factor 77 into prime numbers. What are the prime factors of 77?

See Solution

Problem 656

Find the LCM of 14 and 15 using their prime factors: $14 = 2 \times 7$ and $15 = 3 \times 5$ .

See Solution

Problem 657

Find all positive integers $\leq 100$ with exactly 3 factors.

See Solution

Problem 658

Identify all positive integers up to 100 with exactly three factors. Justify your findings.

See Solution

Problem 659

Describe the pattern and find all positive integers with exactly three factors.

See Solution

Problem 660

Identify all positive integers up to 100 with exactly four factors. Justify your findings.

See Solution

Problem 661

Find a prime and a composite number under 20 that could represent the average weight of a Lakeland Terrier. Explain why.

See Solution

Problem 662

Find the prime factorization of 2,800 using a factor tree and explain each step to a classmate.

See Solution

Problem 663

Find the prime factors of 175 with a factor tree and calculate $2 \times 175$ .

See Solution

Problem 664

Mark the prime numbers from this list: 5, 6, 10, 15, 24, 27, None of the above.

See Solution

Problem 665

Find the least common multiple of the numbers 27, 45, 81, and 225. What is it?

See Solution

Problem 666

Find the prime factors of 140 in ascending order (1 is not prime).

See Solution

Problem 667

Find the prime factors of 99 in order. Remember, 1 is not prime.
$[?] \times [] \times []$

See Solution

Problem 668

Find the prime factors of 140 in ascending order. Remember, 1 is not prime.
$[?] \times[] \times[] \times[]$

See Solution

Problem 669

Find the prime factorization of 140. List factors in order: $[?] \times[] \times[] \times[]$ . Enter the green box number.

See Solution

Problem 670

Find two numbers whose HCF is 5 and LCM is a multiple of 12.

See Solution

Problem 671

Can every number be expressed as a power of another number? Justify your answer.

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Problem 672

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 673

Identify the number that is NOT a prime from the options: A. 37, B. 29, C. 57, D. 83.

See Solution

Problem 674

In the number $123,456,789,000$ , what does the digit 2 represent? A) 2 billion B) 20 billion C) 200 billion D) 2000 billion

See Solution

Problem 675

What is the prime factorization of 520? A. $2^{2} \cdot 5 \cdot 13$ B. $2^{2} \cdot 5^{2} \cdot 13$ C. $2^{3} \cdot 5 \cdot 13$ D. $2 \cdot 5 \cdot 13$

See Solution

Problem 676

Find the prime factorization of 520. Choose from: A. $2^{2} \cdot 5 \cdot 13$ , B. $2^{2} \cdot 5^{2} \cdot 13$ , C. $2^{3} \cdot 5 \cdot 13$ , D. $2 \cdot 5 \cdot 13$ .

See Solution

Problem 677

Factor 300 into its prime components.

See Solution

Problem 678

Find the prime decomposition of 208 in the form $2^{a} \times b$ and determine the values of $a$ and $b$ .

See Solution

Problem 679

Identify the three numbers among these that have exactly two factors: $2, 4, 7, 11, 15$ .

See Solution

Problem 680

Determine the HCF of 70 and 385 using prime factor trees. Prime factors: 385 = 5, 7, 11; 70 = 2, 5, 7.

See Solution

Problem 681

Find the LCM of 154 (factors: 2, 7, 11) and 273 (factors: 3, 7, 13).

See Solution

Problem 682

¿Cuántos paquetes idénticos puede armar Marcus con 36 jabones y 24 cepillos de dientes? Resuelve usando el M.C.D.

See Solution

Problem 683

Marcus tiene 36 jabones y 24 cepillos. ¿Cuántos paquetes idénticos puede armar y cuántos de cada artículo hay en ellos?

See Solution

Problem 684

Find the prime factorization of 196.

See Solution

Problem 685

Determine if the number 25 is prime or composite and provide an explanation.

See Solution

Problem 686

Which pair of numbers is a factor pair of 100? (A) 25 and 75 (B) 3 and 30 (C) 4 and 20 (D) 2 and 50.

See Solution

Problem 687

List all common factors for these pairs: 6. 12 and 30, 7. 12 and 25, 8. 24 and 45.

See Solution

Problem 688

Find the greatest number of scouts per column for troops with 12, 24, and 18 members. Options: A) 9 B) 6 C) 8 D) 4.

See Solution

Problem 689

Solve $-6 x = 66$ . Find $x$ . Also, determine the GCF of 93 and 51.

See Solution

Problem 690

Identify if each number is prime or composite and list its factors: 12, 17, 29, 27, 58.

See Solution

Problem 691

Find the least common multiple (LCM) of the numbers 4, 6, and 10.

See Solution

Problem 692

Find the prime decomposition of 192 in the form $2^{a} \times b$ and determine the values of $a$ and $b$ .

See Solution

Problem 693

Is 125 a perfect square?

See Solution

Problem 694

Are all factors of 16 also factors of its multiples, like 32 and 64? Explain your reasoning.

See Solution

Problem 695

Hugo claims some whole numbers have no common factors, while Jasmine says they always do. Who is right? Explain.

See Solution

Problem 696

Malik has 6 blue and 15 red cubes. How can he share them equally with friends? List ways for both colors and explain the factors.

See Solution

Problem 697

Calculate $\Phi(7) + \Phi(9) + \Phi(12) - \Phi(15)$ . What is the result? a) 6 b) 7 c) 8 d) 9 e) 10

See Solution

Problem 698

Find the probability that a randomly chosen number from $\{n \in \mathbb{N} \mid 51 \leq n \leq 91\}$ is prime. Options: a) $\frac{9}{41}$ b) $\frac{10}{41}$ c) $\frac{11}{41}$ d) $\frac{12}{41}$ e) $\frac{13}{41}$

See Solution

Problem 699

Find the prime factorization of 2646 using exponents.

See Solution

Problem 700

Determine if the following numbers are prime or composite: 81, 43, 72, 93, 53, 87, 13, 27, 88, 19, 69, 7.

See Solution

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

Number Theory

Problem 601

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

Problem 602

Find the highest common factor (HCF) of 70 and 385 using their prime factor trees: 70 = 2×5×72 \times 5 \times 72×5×7, 385 = 5×7×115 \times 7 \times 115×7×11.

Problem 603

Find the LCM of 120 (23×3×52^{3} \times 3 \times 523×3×5) and 198 (2×32×112 \times 3^{2} \times 112×32×11).

Problem 604

Find the unknown unit symbol for the number of oxygen atoms in a NO2\mathrm{NO}_{2}NO2​ molecule: amount or number of 1 one a or nnn.

Problem 605

Which sets N,Z,Q,RN, Z, Q, RN,Z,Q,R does 5\sqrt{5}5​ belong to? Options: Only QQQ, Only QQQ and RRR, Only RRR, Only ZZZ.

Problem 606

Identify the true statement among these: 1. An integer not a natural number exists.2. All real numbers are rational.3. Some rational numbers aren't real.4. All integers except 0 are natural.

Problem 607

Find the prime factorization of the number 32.

Problem 608

Find the prime decomposition of 224 in the form 2a×b2^{a} \times b2a×b and determine the values of aaa and bbb.

Problem 609

Which statement is true? A. Even + 3 is even. B. Even + 3 is prime. C. Even + 3 is odd. D. Prime + 3 is even.

Problem 610

Write 504 as a product of its prime factors in the form p1e1×p2e2×…p_1^{e_1} \times p_2^{e_2} \times \ldotsp1e1​​×p2e2​​×…

Problem 611

Find the prime factorization of 252 in index form and identify which two of the following are factors of 252: 1) 22=2×1122=2 \times 1122=2×11 2) 24=23×324=2^{3} \times 324=23×3 3) 9=329=3^{2}9=32 4) 14=2×714=2 \times 714=2×7 5) 49=7249=7^{2}49=72.

Problem 612

Which numbers are factors of 25,704, given its prime factorization 23×33×7×172^{3} \times 3^{3} \times 7 \times 1723×33×7×17? Check all that apply: 63: 32×73^{2} \times 732×7, 513: 33×193^{3} \times 1933×19, 34: 2×172 \times 172×17, 8: 232^{3}23, 16: 242^{4}24.

Problem 613

Find the HCF of 60 and 114.

Problem 614

Find the GCD of 36 and 55 using the Euclidean algorithm, then express it as an integral combination and find the inverse of 36 mod 5536 \bmod 5536mod55.

Problem 615

Find the smallest composite number greater than 1 that is not divisible by 222, 333, 555, or 777.

Problem 616

Count how many prime numbers can be formed using the digits 1 to nnn (where nnn is from 1 to 9) exactly once.

Problem 617

Find a positive integer with 8 divisors, where two divisors are 21 and 33.

Problem 618

Determine if the following numbers are prime or composite: 44, 12, 50, 83, 27, 53, 24, 67, 78.

Problem 619

Determine if the following numbers are prime or composite: 44, 12, 50, 83, 53, 24, 34, 27, 67, 78.

Problem 620

Find the LCM and HCF of G=25×33×5\mathrm{G}=2^{5} \times 3^{3} \times 5G=25×33×5 and H=22×36×5×7\mathrm{H}=2^{2} \times 3^{6} \times 5 \times 7H=22×36×5×7.

Problem 621

Find common multiples of 60 (22×3×52^{2} \times 3 \times 522×3×5) and 1120 (25×5×72^{5} \times 5 \times 725×5×7) from the options.

Problem 622

Find the common multiples of 60 (22×3×52^{2} \times 3 \times 522×3×5) and 2240 (26×5×72^{6} \times 5 \times 726×5×7) from the options.

Problem 623

Find the prime factors of 52. The prime factorization is 22×132^2 \times 1322×13.

Problem 624

Find the prime factors of 102. The prime factorization is 2×3×172 \times 3 \times 172×3×17.

Problem 625

Express 52 as a product of its prime factors. 52= 52 = 52=

Problem 626

Find the prime factorization of 64.

Problem 627

Find the greatest length in meters for equal pieces from 44 m blue ribbon and 110 m red ribbon with no leftover.

Problem 628

Find the GCF of 62 and 70 using prime factorization. Show your work for full credit.

Problem 629

Find the prime factorization of 145.

Problem 630

Czy 234+3562^{34}+3^{56}234+356 jest liczbą pierwszą? Uzasadnij swoją odpowiedź.

Problem 631

Find the number of factors of 40.

Problem 632

Identify the relatively prime pair: A) 5 and 75 B) 12 and 51 C) 7 and 20 D) 15 and 21

Problem 633

Identify the pair of numbers that are relatively prime: A) 119 & 209, B) 88 & 121, C) 144 & 165, D) 42 & 385.

Problem 634

Find the common primes in the factor trees of 30 and 42, then determine their HCF as a product of primes and its value.

Problem 635

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

Problem 636

Find the HCF of 70 and 385 using prime factor trees: 70=2×5×770 = 2 \times 5 \times 770=2×5×7, 385=5×7×11385 = 5 \times 7 \times 11385=5×7×11.

Problem 637

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

Problem 638

Identify the common prime factors of 6435 and 8190 from the options below: 2 × 3 × 5 × 7 × 11 × 13, 3 × 13, 2 × 3, 2 × 3^{4} × 5^{2} × 7 × 11 × 13^{2}, 3^{2} × 5 × 13, 2 × 7 × 11, 3^{2}.

Problem 639

Problem 640

Find the prime factors of 105 using a tree and then calculate the HCF of 30 and 105.

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

Find the LCM of 120 ( $2^{3} \times 3 \times 5$ ) and 198 ( $2 \times 3^{2} \times 11$ ).

Find the unknown unit symbol for the number of oxygen atoms in a $\mathrm{NO}_{2}$ molecule: amount or number of 1 one a or $n$ .

Which sets $N, Z, Q, R$ does $\sqrt{5}$ belong to? Options: Only $Q$ , Only $Q$ and $R$ , Only $R$ , Only $Z$ .

Identify the true statement among these:
1. An integer not a natural number exists.
2. All real numbers are rational.
3. Some rational numbers aren't real.
4. All integers except 0 are natural.

Find the prime decomposition of 224 in the form $2^{a} \times b$ and determine the values of $a$ and $b$ .

Write 504 as a product of its prime factors in the form $p_1^{e_1} \times p_2^{e_2} \times \ldots$

Find the prime factorization of 252 in index form and identify which two of the following are factors of 252:
1) $22=2 \times 11$ 2) $24=2^{3} \times 3$ 3) $9=3^{2}$ 4) $14=2 \times 7$ 5) $49=7^{2}$ .

Which numbers are factors of 25,704, given its prime factorization $2^{3} \times 3^{3} \times 7 \times 17$ ? Check all that apply:
63: $3^{2} \times 7$ , 513: $3^{3} \times 19$ , 34: $2 \times 17$ , 8: $2^{3}$ , 16: $2^{4}$ .

Find the GCD of 36 and 55 using the Euclidean algorithm, then express it as an integral combination and find the inverse of $36 \bmod 55$ .

Find the smallest composite number greater than 1 that is not divisible by $2$ , $3$ , $5$ , or $7$ .

Count how many prime numbers can be formed using the digits 1 to $n$ (where $n$ is from 1 to 9) exactly once.

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

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

Find the common multiples of 60 ( $2^{2} \times 3 \times 5$ ) and 2240 ( $2^{6} \times 5 \times 7$ ) from the options.

Find the prime factors of 52. The prime factorization is $2^2 \times 13$ .

Find the prime factors of 102. The prime factorization is $2 \times 3 \times 17$ .

Express 52 as a product of its prime factors. $52 =$

Czy $2^{34}+3^{56}$ jest liczbą pierwszą? Uzasadnij swoją odpowiedź.

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

Identify the correct prime factorization of 200 in exponential form: $2^{3} \cdot 5^{2}$ , $2 \cdot 10^{2}$ , $2^{2} \cdot 5^{2}$ , or $8 \cdot 25$ .

8是一个完全平方数吗？是否重看提交

Find the prime factorization of 14 and list the factors in ascending order, like $2 \times 7$ .

Why is there a 3 outside the cube root in $\sqrt[3]{54} = 3 \sqrt[3]{2}$ ? Explain its significance.

Determine the prime factors of $50,225,675$ .

Which of these numbers is NOT a multiple of $17$ : 17, 68, 117, 1700, or 3434?

Optimize code that checks if $num1$ is divisible by $num2$ or both are zero. Discuss division, modulo, and conditionals.

20. (a) Define what it means for an integer $a$ to divide another integer $b$ . (b) Prove that $7$ divides $12^{n}-5^{n}$ for all $n \in \mathbf{N}$ .

Find the LCM of 14 and 15 using their prime factors: $14 = 2 \times 7$ and $15 = 3 \times 5$ .

Find all positive integers $\leq 100$ with exactly 3 factors.

Find the prime factors of 175 with a factor tree and calculate $2 \times 175$ .

Find the prime factors of 99 in order. Remember, 1 is not prime.
$[?] \times [] \times []$

Find the prime factors of 140 in ascending order. Remember, 1 is not prime.
$[?] \times[] \times[] \times[]$

Find the prime factorization of 140. List factors in order: $[?] \times[] \times[] \times[]$ . Enter the green box number.

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

In the number $123,456,789,000$ , what does the digit 2 represent? A) 2 billion B) 20 billion C) 200 billion D) 2000 billion