CAT 2024 - Slot 2 Quant Question 5
This is a detailed step-by-step solution for CAT 2024 - Slot 2 Quant Question 5. Understand the concept, common mistakes, and expert tips to solve similar questions in CAT exam.
CAT 2024 Slot 2 - Quant
Looking at the additional information about the prime numbers should make one realise that they are the key to solving the question.
f(16000) can be written as f(28 × 54)
Now, we can try to find these individual values:
For any prime p: f(p) = 1
f(p2) = f(p) f(p) + f(p) + f(p) = 1 + 1 + 1 = 3
f(p3) = f(p2) f(p) + f(p2) + f(p) = 3 + 3 + 1 = 7
This way, we can find the function output for any prime number raised to a power.
We can see that each new exponent is twice the previous output +1, solving this way till prime raised to power 8
f(p4) = 7 + 7 + 1 = 15
f(p5) = 15 + 15 + 1 = 31
f(p6) = 31 + 31 + 1 = 63
f(p7) = 63 + 63 + 1 = 127
f(p8) = 127 + 127 + 1 = 255
Using these values in the original expression of f(28 × 54) = f(28) f(54) + f(28) + f(54) we get
f(28 × 54) = (255 × 15) + 255 + 15 = 4095