To find the smallest number divisible by both 12 and 15, we need to calculate the least common multiple (LCM). The prime factorizations are: 12 = 2² × 3 15 = 3 × 5 LCM = 2² × 3 × 5 = 60. The sum of the digits of 60 is 6 + 0 = 6.

To check divisibility by 9, sum the digits of the number: 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45. Since 45 ÷ 9 = 5, the remainder is 0.

The sequence is obtained by adding 3 to the previous term.

The sum of the digits of 56789 is: 5 + 6 + 7 + 8 + 9 = 35. Now, divide 35 by 9: 35 ÷ 9 = 3 remainder 8.

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

(2³)² = 2⁶, (3²)³ = 3⁶, (4⁴) = (2²)⁴ = 2⁸. So, (2³)² × (3²)³ ÷ (4⁴) = 2⁶ × 3⁶ ÷ 2⁸ = 3⁶ × 2⁶ ÷ 2⁸ = 3⁶ ÷ 2² = 729 ÷ 4 = 182.25.

Let the original number be 10a + b, where a and b are the tens and ones digits respectively. The sum of the digits is a + b = 10, and when the digits are reversed, the new number is 10b + a. We are given that 10b + a = 10a + b + 9. Simplifying: 10b + a = 10a + b + 9 9b – 9a = 9 b – a = 1 Thus, b = a + 1. Substitute into the sum equation: a + (a + 1) = 10, 2a + 1 = 10, 2a = 9, a = 4.5 (not an integer, so no solution).

To check divisibility by 9, sum the digits of the number: 1 + 0 + 0 + 1 + 0 + 0 + 1 + 0 + 0 + 1 = 5. Now divide 5 by 9: 5 ÷ 9 gives a remainder of 5.

The LCM of 8 and 12 is 24.

Using the divisibility rule for 11, alternate the sum and difference of the digits: 5 – 6 + 7 – 8 + 9 = 7. Now divide 7 by 11: 7 ÷ 11 gives a remainder of 7.

The prime factorization of 300 is: 300 = 2² × 3 × 5². The number of prime factors is 3 (2, 3, and 5).

To check divisibility by 3, add the digits of the number: 7 + 6 + 5 = 18. Since 18 is divisible by 3, 765 is divisible by 3.

The sum of the first 10 odd numbers is 1 + 3 + 5 + … + 19 = 100.

Start by dividing 84 by the smallest prime (2): 84 ÷ 2 = 42, 42 ÷ 2 = 21. 21 is not divisible by 2, so divide by 3: 21 ÷ 3 = 7. 7 is a prime number. So, the prime factorization of 84 is 2² × 3 × 7.

Using the laws of exponents: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625.

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.

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.

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

3³ × 2³ ÷ 6³ = (3 × 2)³ ÷ 6³ = 6³ ÷ 6³ = 1.

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