We know the remainder when the number is divided by 9 is 5, which means the number can be expressed as 9k + 5 for some integer k. Now, when this number is divided by 3, we have: 9k + 5 = (3 × 3k) + 5 Since 9k is divisible by 3, the remainder is the same as the remainder of 5 divided by 3. 5 ÷ 3 gives a remainder of 2.

Using the laws of exponents: 10⁴ ÷ 10² = 10⁴⁻² = 10² = 100.

The prime factorization of 420 is: 420 = 2² × 3 × 5 × 7. Thus, the largest prime factor is 7.

The perfect squares among the options are: 361 = 19². Therefore, 361 is the perfect square.

To find the smallest number divisible by 10, 15, and 20, we find their Least Common Multiple (LCM). Prime factorizations: 10 = 2 × 5, 15 = 3 × 5, 20 = 2² × 5. LCM = 2² × 3 × 5 = 60.

The prime factorizations of 72 and 120 are: 72 = 2³ × 3² 120 = 2³ × 3 × 5 The GCD is obtained by taking the lowest powers of the common prime factors: GCD = 2³ × 3 = 24.

First, multiply 7 × 11 × 13 = 1001. Now divide by 7: 1001 ÷ 7 = 143.

The sequence is obtained by multiplying the previous term by 2.

To convert 45 to binary, divide by 2 and record the remainders: 45 ÷ 2 = 22, remainder 1 22 ÷ 2 = 11, remainder 0 11 ÷ 2 = 5, remainder 1 5 ÷ 2 = 2, remainder 1 2 ÷ 2 = 1, remainder 0 1 ÷ 2 = 0, remainder 1 Reading the remainders from bottom to top, we get 101101.

The prime factorization of 12 is: 12 = 2² × 3, The prime factorization of 18 is: 18 = 2 × 3². The LCM is the product of the highest powers of all primes: LCM = 2² × 3² = 36.

For a number to be divisible by 6, it must be divisible by both 2 and 3. Thus, the number must be even (divisible by 2) and the sum of its digits must be divisible by 3.

12345 ÷ 7 = 1763 remainder 4. So, the remainder is 4.

Solving the equation x/4 = 9 gives x = 36.

Apply the divisibility rule for 11: Sum the alternating digits: 9 – 8 + 7 – 6 + 5 – 4 + 3 – 2 + 1 = 5. Now divide 5 by 11: 5 ÷ 11 gives a remainder of 5.

The smallest such number is 50. 50 can be written as: 50 = 1² + 7², 50 = 5² + 5².

The prime factorization of 144 is: 144 = 2⁴ × 3². The largest prime factor is 3.

For a number to be divisible by both 6 and 9, it must be divisible by their Least Common Multiple (LCM), which is 18. Among the options, only 54 is divisible by 18.

The prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. There are 15 prime numbers in total between 1 and 50.

The first 15 square numbers are: 1², 2², 3², 4², 5², 6², 7², 8², 9², 10², 11², 12², 13², 14², 15². The sum of these is: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 + 169 + 196 + 225 = 1225.

The LCM of 6 and 9 is 18.