Number System

Concept: The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more integers is the largest positive integer that divides each of the integers without a remainder. We can find the HCF using prime factorization or the Euclidean algorithm. Method 1: Prime Factorization Step 1: Find the prime factorization of 96 Step 2: Find the prime factorization of 404 Since 101 is a prime number (it is not divisible by any prime numbers less than or equal to , i.e., 2, 3, 5, 7), Step 3: Identify common prime factors and their lowest powers The common prime factor is 2. The lowest power of 2 present in both factorizations is . (3 is a factor of 96 but not 404. 101 is a factor of 404 but not 96). Step 4: Calculate the HCF HCF = Product of common prime factors raised to their lowest powers. HCF = . Method 2: Euclidean Algorithm Step 1: Divide the larger number by the smaller number and find the remainder. (Since , remainder is ) Step 2: Replace the larger number with the smaller number and the smaller number with the remainder, and repeat the division. Now divide 96 by 20. Step 3: Repeat the process. Now divide 20 by 16. Step 4: Repeat the process. Now divide 16 by 4. The last non-zero remainder is the HCF. HCF = 4. Both methods give HCF = 4. This matches option (2).

Concept: A rational number (a fraction where and are integers and ) has a terminating decimal expansion if and only if its denominator , when the fraction is in its simplest form, has prime factors only of 2 and/or 5. The number of decimal places after which it terminates is determined by the highest power of 2 or 5 in the prime factorization of the denominator of the simplified fraction. Step 1: Simplify the given fraction The given fraction is . First, find the greatest common divisor (GCD) of the numerator and the denominator. Prime factorization of 147: . Prime factorization of 120: . The GCD of 147 and 120 is 3. Divide both numerator and denominator by their GCD (3): So, the simplified fraction is . Step 2: Prime factorize the denominator of the simplified fraction The denominator of the simplified fraction is 40. Prime factorization of 40: . Step 3: Determine if the decimal expansion terminates Since the prime factors of the denominator (40) are only 2 and 5 (specifically and ), the decimal expansion of will terminate. Step 4: Determine the number of decimal places The decimal expansion will terminate after a number of places equal to the highest power of 2 or 5 in the prime factorization of the denominator. The powers are:
Power of 2 is 3 (from ).
Power of 5 is 1 (from ). The highest of these powers is 3. Therefore, the decimal expansion of will terminate after 3 decimal places. To verify by actual division: . The decimal 1.225 has 3 places after the decimal point. This matches option (3).

Concept: To simplify expressions involving square roots (surds), we look for perfect square factors within the numbers under the square root. The property is used. Step 1: Simplify each square root term



Step 2: Substitute the simplified terms back into the expression The expression is . Substituting the simplified terms: Step 3: Factor out common terms from the numerator and denominator
Numerator:
Denominator: So the expression becomes: Step 4: Cancel out the common factor Since is a common factor in both the numerator and the denominator, and it is not zero, we can cancel it out: Step 5: Calculate the final value The value of the expression is 2. This matches option (2).