Professor Nitwit obtains a hash number of a given positive integer greater…
2024
Professor Nitwit obtains a hash number of a given positive integer greater than 3 as follows. He subtracts 2 from the number (to get the new number), and multiplies the new number by 2 to get a term. He repeats this with the new number (to get newer numbers and terms) until the number becomes 2 or 1. The hash is defined as the sum of all the numbers generated in the process.
For example, with the number 5, he multiplies (5 - 2 = 3) by 2 to get the first term 6. He multiplies (3 - 2 = 1) by 2 to get the second term 2. As the number has become 1, he stops. The hash is the sum of the two numbers (6 + 2) = 8.
If Professor Nitwit is given 3 numbers 4, 9 and 13, what is the sum of the hash numbers he obtains for the three numbers?
Answer: D. 108 — Repeatedly subtracting a fixed number 2 from a value generates numbers in an arithmetic progression (AP) with common difference 2. Professor Nitwit's hash…
- A.
250
- B.
150
- C.
200
- D.
108
Attempted by 41 students.
Show answer & explanation
Correct answer: D
Repeatedly subtracting a fixed number 2 from a value generates numbers in an arithmetic progression (AP) with common difference 2. Professor Nitwit's hash number for an input n is twice the sum of this AP: hash(n) = 2 x (sum of n-2, n-4, n-6, ... down to the last positive term before the process stops at 1 or 2).
For n = 4: the process gives (4 - 2) = 2, and it stops immediately since 2 is a stopping value. Sum of this one-term AP = 2, so hash(4) = 2 x 2 = 4.
For n = 9: the process gives 7, 5, 3, 1 (it stops at 1). Sum of this AP = 7 + 5 + 3 + 1 = 16, so hash(9) = 2 x 16 = 32.
For n = 13: the process gives 11, 9, 7, 5, 3, 1 (it stops at 1). Sum of this AP = 11 + 9 + 7 + 5 + 3 + 1 = 36, so hash(13) = 2 x 36 = 72.
Cross-check with the worked example already given in the question: for n = 5, the process gives 3, 1, so hash(5) = 2 x (3 + 1) = 8, exactly matching the stem and confirming the method.
Adding the three hash numbers: hash(4) + hash(9) + hash(13) = 4 + 32 + 72 = 108.