Arun Sharma Test: Number System- 2 - CAT MCQ

Given that (10²⁵ - 7) - (10²⁴ + x) is divisible by 3:

• Start with the expression: 10²⁵ - 7 - 10²⁴ - x.
• Factor out 10²⁴:
  • Resulting in: 10²⁴(10 - x) - 7.
• The simplified form is: 9 × 10²⁴ - 7 - x.

For the expression to be divisible by 3, we find:

• Setting 9 × 10²⁴ - 7 - x equal to 0 modulo 3.
• We can derive that the value of x = 2.

Therefore, the correct option is B.

=x                                     
All the digits written once = 1762125.125 = 10000x
All the digits without Bar written once = 1762.125 = 10x
Rational form = 

► There are two possibilities for a
► Case 1: a is even

⇒ 6a  = Even x Even = Even
⇒ 6a + 12 = Even + Even = Even

► Case 2: a is odd

⇒ 6a = Even x Odd = Even
⇒ 6a + 12 = Even + Even = Even

► In both the cases, the answer is even, but the question says its odd. so, the information provided is inconsistent.

• Focus on last-two-digit parts: 22, 23, 25, 27, 29.

• Multiply stepwise, keeping last two digits each time:

  • 22×23 = 506 → 06

  • 06×25 = 150 → 50

  • 50×27 = 1,350 → 50

  • 50×29 = 1,450 → 50
    Hence the product ends with 50.

► For abc to be Odd the only case possible is none of them being Even i.e. a, b and c all three are Odd

⇒ a² + b² + c²
⇒ Odd² + Odd² + Odd²
⇒ Odd + Odd + Odd
⇒ Odd

Vice versa can also be true

► There is only one Even prime number i.e. 2. 

► We know the property that a prime number when divided by 6 gives a remainder of 1 or 5.
► Option A : 8831 divided by 6 gives a remainder of 5
► Option B : 8829 divided by 6 gives a remainder of 3
► Option C : 8833 divided by 6 gives a remainder of 1, but is divisible by 11
► Option D : 8835 divided by 6 gives a remainder of 3

► Option A : a² + b² = Odd² + Odd² = Odd + Odd = Even
► Option B  : a² + 2b² = Odd² + Even × Odd² = Odd + Even = Odd
► Option C ∶ 2a² + b² = Even × Odd² + Odd² = Even + odd = Odd
► Option D ∶ a² + b² + ab = Odd² + Odd² + Odd × Odd = Odd + Odd + Odd = Odd

Step 1 – Formula for LCM of fractions
For fractions n1/d1, n2/d2, … in lowest form:

LCM = (LCM of numerators) ÷ (GCD of denominators)

Step 2 – Write fractions clearly
The given fractions are 5/2, 8/9, 11/14.
They are already in simplest form.

Step 3 – Find LCM of numerators
Numerators are 5, 8, 11.

• Prime factors: 5, (8 = 2³), 11

• LCM = 2³ × 5 × 11 = 8 × 55 = 440

Step 4 – Find GCD of denominators
Denominators are 2, 9, 14.

• GCD(2, 9) = 1

• GCD(1, 14) = 1
  So GCD = 1

Step 5 – Find LCM of fractions
LCM = 440 ÷ 1 = 440

Step 1 – Rule
The GCD is found by taking the common prime factors with their smallest exponents.

Step 2 – Compare prime factors

• For 2 → exponents are 3 (in P) and 5 (in Q). Take the smaller one = 2³.

• For 3 → exponents are 10 (in P) and 1 (in Q). Take the smaller one = 3¹.

• For 5 → only in P, not in Q → ignore.

• For 7 → only in Q, not in P → ignore.

Step 3 – Multiply
GCD = 2³ × 3 = 8 × 3 = 24 = 2³.3
 

Step 1 – Recall the definition of a prime number
A prime number is a number greater than 1 that has exactly two positive divisors: 1 and itself.

Step 2 – Check divisibility by 17
If a prime number is divisible by 17, then it must be 17 itself, because any other multiple of 17 (like 34, 51, 68…) has more than two divisors, so it is not prime.

Step 3 – Check numbers less than 1000

• The only prime number divisible by 17 is 17 itself.

Step 1. Prime factorization of 1420

1420 ÷ 2 = 710
710 ÷ 2 = 355
355 ÷ 5 = 71
71 is prime.

So,
1420 = 2² × 5 × 71

Step 2. Divisor formula

If N = p₁^a × p₂^b × p₃^c …,
then number of divisors = (a+1)(b+1)(c+1)…

Here, 1420 = 2² × 5¹ × 71¹
So divisors = (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12

► Being Prime numbers any number would be divisible by 1 or itself.
► Since 16 is not a prime number. There would be no other prime number divisible by 16. 

720= 2⁴ × 3² × 5¹. Number of factors = 5 × 3 × 2 = 30. Option (d) is correct.

tep 1 – Factor each number into primes

• 15 = 3 × 5 → 15³ = 3³ × 5³

• 21 = 3 × 7 → 21² = 3² × 7²

• 35 = 5 × 7 → 35² = 5² × 7²

• 3⁴ stays as 3⁴

Step 2 – Write entire expression in prime factors

Numerator: 15³ × 21² = (3³ × 5³) × (3² × 7²) = 3⁵ × 5³ × 7²
Denominator: 35² × 3⁴ = (5² × 7²) × 3⁴ = 3⁴ × 5² × 7²

Step 3 – Simplify by subtracting exponents

• 3: 3⁵ ÷ 3⁴ = 3¹ = 3

• 5: 5³ ÷ 5² = 5¹ = 5

• 7: 7² ÷ 7² = 1

So the expression simplifies to 3 × 5 = 15