Option A is the correct answer.

Explanation: The divisibility rule of 11 is that it must be subtracted and then added to the digits in an alternating pattern from left to right. If the answer is 0 or 11 then it is divisible by 11.

Hence,

If P = 9

(9 + 1) - (7 + 3) = 10 - 10 = 0

Therefore, P = 9

Option D is the correct answer.

Explanation: Dividend = Divisor × Quotient + Remainder

⇒ 14528 = Divisor × 83 + 3

⇒ 14528 - 3 = Divisor × 83

⇒ 14525/83 = Divisor

⇒ Divisor = 175

∴ Divisor = 175

The lowest 3-digit no. is 100 and the highest is 999.

The lowest 3-digit multiple of 9 is 108, and the highest is 999.

No. of multiples in between = 999−1089+1=100

: HCF × LCM = n1 × n2; n1 = number1, n2 = number2

Let x be the other number.

210 × 35 = 105 × x

⇒ x = 70

Let 4¹ = 4, when divided by 3 gives the remainder 1

⇒ Four power odd number when divided by 3 gives the remainder 1

Similarly, 4³ = 64, when divided by 3 gives the remainder 1

∴ 4¹³ divided by 3 also gives the remainder 1

LCM of 10, 17 = 170

We have to find the number of multiples of 170 < 1000

Number of multiples = integer value of (1000/170) = integer part of 5.88 = 5

∴ there are 5 multiples of both 10 and 17 less than 1000 are 5

Factors of 99 = 1, 3, 9, 11, 33, 99

Factors of 101 = 1, 101

Factors of 176 = 1, 2, 4, 8, 11, 16, 22, 44, 88, 176

Factors of 182 = 1, 2, 7, 13, 14, 26, 91, 182

Thus 176 has the maximum number of factors

Let the number be ‘x’

x = 6y + 3

x² = (6y + 3)²

x² = 36y² + 36y + 9

x² = 6 × (6y² + 6y + 1) + 3

∴ the square of the same number is divided by 6, the remainder will be 3.

: Since the numbers are co-prime, they contain only 1 as the common factor

Also, the given two products have the middle number in common

So, middle number

H.C.F. of 551 and 1073 = 29

First number = 551/29 = 19

Third number = 1073/29 = 37

Required sum = (19 + 29 + 37) = 85

∴ Sum of the three numbers is 85

Let’s assume the 2 consecutive natural numbers be ‘n’ and ‘n + 1’

⇒ n² + (n + 1)² = 145

⇒ 2n² + 1 + 2n = 145

⇒ 2n² + 2n - 144 = 0

⇒ n² + n - 72 = 0

By middle term splitting, we get,

n² + 9n - 8n - 72 = 0

n (n + 9) - 8(n + 9) = 0

(n - 8)(n + 9) = 0

n = 8, -9

n can’t take negative values, as it is a natural number.

n = 8

n + 1 = 9

∴  Two numbers are 8 and 9

Sum of squares of 15 and 14 = 15² + 14² = 225 + 196 = 421

The nearest perfect square is 21² (= 441).

Smallest number that should be added the sum of squares of 15 and 14, so that the resulting number is a perfect square = 441 – 421 = 20

∴ Smallest number that should be added the sum of squares of 15 and 14, so that the resulting number is a perfect square = 20