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.

Therefore, the last two digits of the multiplication 122×123×125×127×129 are: 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

LCM of 5/2, 8/9 and 11/14 would be given by: (LCM of numerators)/(HCF of denominators) =440/1 =440

For the GCD, take the least powers of all common prime factors.

► Being Prime numbers any number would be divisible by 1 or itself.
► Since 17 is a prime number. It will be the only prime number less than 1000 which will be divisible by 17.

1420=142 × 10=2²×7¹¹×5¹.
Thus, the number of factors of the number would be (2+1)(1+1)(1+1)=3×2×2=12.
Option (d) is correct.

► 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.

The given expression can be written as: 5³ × 3³ × 3² × 7²/5² × 7² × 3⁴ = 5³ × 3⁵ × 7²/5² × 7² × 3⁴ = 15.
Option (b) is correct.