Consider a 2-digit number N. Let P be the product of the digits of the number. If P is added to square of the digit in the tens place of N, we get 84. If P is added to the square of the digit in the unit place of N, we get 60. What is the value of P + N ?

Given:

Let the 2-digit number be N = 10×a + b, where:

a = digit in tens place

b = digit in units place

P = a × b

Formula used:

Condition 1: P + a² = 84

Condition 2: P + b² = 60

Calculations:

From (1): P = 84 − a²

From (2): P = 60 − b²

⇒ 84 − a² = 60 − b²

⇒ 24 = a² − b²

⇒ (a − b)(a + b) = 24

Try factor pairs of 24: (1,24), (2,12), (3,8), (4,6)

Try a − b = 2 and a + b = 12

⇒ Add: 2a = 14 ⇒ a = 7, b = 5

Check P = a × b = 7 × 5 = 35

P + a² = 35 + 49 = 84 

P + b² = 35 + 25 = 60

N = 10×7 + 5 = 75

P = 35

P + N = 35 + 75 = 110

∴ The correct answer is: 110