Govt. Exams
Entrance Exams
12 = 2²×3, 18 = 2×3². LCM = 2²×3² = 4×9 = 36
Sum of first n natural numbers = n(n+1)/2. For n=15: 15×16/2 = 240/2 = 120
# 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.
Digit sum = 9 + 9 + 9 + 9 = 36. Note: A number and its digit sum have the same remainder when divided by 9.
72 = 2³ × 3². Number of divisors = (3+1)(2+1) = 4 × 3 = 12.
61 is only divisible by 1 and itself. 51 = 3 × 17, 57 = 3 × 19, 63 = 9 × 7 are composite numbers.
12 = 2² × 3; 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.
Find HCF of 48 and 64 using Euclidean algorithm: 64 = 48 × 1 + 16; 48 = 16 × 3 + 0. Therefore, HCF = 16.
Prime factorization of 360: 360 = 2³ × 3² × 5.
The unique prime factors are 2, 3, and 5.
Product = 2 × 3 × 5 = 30.
Let the number be n.
Given: n = 8k + 5 for some integer k.
When divided by 4: n = 8k + 5 = 4(2k) + 4 + 1 = 4(2k + 1) + 1.
Therefore, remainder = 1.
