72 = 2³ × 3². Number of divisors = (3+1)(2+1) = 4 × 3 = 12.
Digit sum = 9 + 9 + 9 + 9 = 36. Note: A number and its digit sum have the same remainder when divided by 9.
# Solution: Finding the Value of x Using Exponent Laws
To solve exponential equations, express all terms with the same base and apply exponent division rules.
Step 1: Express 216 as a Power of 6
We need to rewrite 216 in terms of base 6 to simplify the equation.
Step 2: Substitute and Apply Division Rule
Substitute this into the original equation and use the rule that \(\frac{a^m}{a^n} = a^{m-n}\).
Step 3: Equate the Exponents
Since the bases are equal, the exponents must be equal.
We are given:
216
6
x
=6
Since:
216=6
3
Substitute:
6
3
6
x
=6
Using laws of exponents:
6
x−3
=6
1
Therefore,
x−3=1
x=4
So, the value of x is:
4
The answer is (C) 4.
Sum of first n natural numbers = n(n+1)/2. For n=15: 15×16/2 = 240/2 = 120
12 = 2²×3, 18 = 2×3². LCM = 2²×3² = 4×9 = 36
For divisibility by 6, number must be divisible by both 2 and 3. 104÷6 = 17.33... (not divisible). Others: 72, 84, 90 are all divisible by 6
Let the number be x. According to the problem: 5x - 3 = 47. Therefore, 5x = 50, so x = 10.
12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 8 × 9 = 72.
We need to find x where x² = 169. Taking square root: x = √169 = 13.
We need a number of the form 8k + 5. Check: 29 = 8(3) + 5 = 24 + 5. ✓
