Number System - Aptitude MCQ Questions and Solutions with Explanations

Let the unit's digit be x and ten's digit be y
Then, the number is 3422213yx
Also, 99 = (11 × 9), where 11 and 9 are co-primes
Since the given number is divisible by 9, it follows that (3 + 4 + 2 + 2 + 2 + 1 + 3 + y + x) = (17 + y + x) must be divisible by 9
So, y + x = 1 or y + x = 10
Again, the given number is divisible by 11
So, (x + 3 + 2 + 2 + 3) - (y + 1 + 2 + 4) = x - y + 3 is either 0 or 11
∴ (x - y + 3 = 0 or x - y + 3 = 11)
⇒ (y - x = 3 or x - y = 8)
Now, (y + x = 1 and y - x = 3)
⇒ y = 2 and x = - 1
(y + x = 1 and x - y = 8)
⇒ x = $$\frac{9}{2}$$
(y + x = 10 and y - x = 3)
⇒ y = $$\frac{13}{2}$$
(y + x = 10 and x - y = 8)
⇒ x = 9 and y = 1
Thus, x = 9, y = 1
So, required number is 342221319

As proved in the above question,
Divisor = 4375 + 2986 - 2361
             = 5000

Sum of digits of numbers from 1 to 10 = 46
Sum of digits of numbers from 11 to 20 = 56
Sum of digits of numbers from 21 to 29 = 63
Sum of digits of the given number = 46 + 56 + 63 = 165
So, the required remainder is the remainder obtained on dividing 165 by 9, which is 3

m³- 3m² + 3m + 3n + 3n² + n³
= - 64 - 48 - 12 - 6 + 12 - 8
= -126

We have 24 = 3 × 8, where 3 and 8 are co-primes
Clearly,
718 is not divisible by 8. So, 35718 is not divisible by 8
810 is not divisible by 8. So, 63810 is not divisible by 8
804 is not divisible by 8. So, 537804 is not divisible by 8
736 is divisible by 8. So, 3125736 is divisible by 8
Also, sum of its digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3
So, 3125736 is divisible by 3 also.
Hence, it is divisible by 24.

Let $$\frac{46351 - 36418 - 4505}{x}$$     = 1357
Then,
x = $$\frac{46351 - (36418 - 4505)}{1357}$$
x = $$\frac{(46351 - 40923)}{1357}$$
x = $$\frac{5428}{1357}$$
x = 4

$$\frac{(a - b)}{3.5}$$ = $$\frac{4}{7}$$
⇒ (a - b) = $$\frac{4}{7}$$ × $$\frac{7}{2}$$ = 2
⇒ b < a

(xⁿ + aⁿ) is divisible by (x + a) when n is odd
∴ (25²⁵ + 1²⁵) is divisible by (25 + 1)
⇒ (25²⁵ + 1) is divisible by 26
⇒ On dividing 25²⁵ by 26, we get (26 - 1) = 25 as remainder

The number of prime numbers from 100 to 200 is 21.

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

Clearly, 100 is divisible by 2. So, 100 is not prime.
(101) < (11)² and prime numbers less than 11 are 2, 3, 5, 7
Clearly, 101 is not divisible by any of 2, 3, 5 and 7
Hence, 101 is the smallest 3-digit prime number.