24Directions: A word and number arrangement machine, when given an input line…
2025
24
Directions: A word and number arrangement machine, when given an input line of numbers, rearranges them following a particular rule in each step. The following is an illustration of an input and rearrangement.
Input: PA683 LN475 TU294 BK861 OR537 MI742 Step I: BO643 KM491 SV812 AJ645 PQ492 JL492 Step II: OB13 MK14 VS11 JA15 QP15 LJ15 Step III: ob26 mk28 vs22 ja30 qp30 lj30 Step IV: na78 lj84 ur66 iz90 po90 ki90 Step V: %#69 %%95 #%77 #%80 %#80 %#80
And Step V is the last step of the rearrangement for the given input. As per the rules followed in the above steps, find out in each of the following questions the appropriate steps for the given input.
Input: DE584 AX739 PM261 WS948 CN375 UO682
How many numbers are divisible by 4 in Step IV?
- A.
One
- B.
Two
- C.
Three
- D.
Four
- E.
More than four
Show answer & explanation
Correct answer: B
Concept
In a machine-input arrangement, each item transforms in place by a fixed rule at every step. For a divisibility question, only the NUMBER attached to each word matters, so we track how that number changes from the Input through the steps.
Reading the worked example, the number of each word changes step-by-step by these rules:
Input to Step I: square the largest digit to form the first two digits, then append the absolute difference of the other two digits.
Step I to Step II: replace the number by its digit sum.
Step II to Step III: double the number.
Step III to Step IV: triple the number.
For example, 683: largest digit 8, so 82 = 64 and |6 - 3| = 3, giving Step I number 643 (digit sum 13, then x2 = 26, then x3 = 78 at Step IV).
Combining the steps, the Step-IV number = 3 x 2 x (digit sum of the Step-I number) = 6 x (Step-II number). A value 6k is a multiple of 4 exactly when k is even. Therefore a Step-IV number is divisible by 4 if and only if its Step-I digit sum is even.
Apply the Input to Step I rule to the new input DE584 AX739 PM261 WS948 CN375 UO682 and test each Step-I digit sum for evenness:
Input number | Step I number | Digit sum (Step II) | Step IV (x6) | Divisible by 4? |
|---|---|---|---|---|
584 | 641 | 11 | 66 | No |
739 | 814 | 13 | 78 | No |
261 | 361 | 10 | 60 | Yes |
948 | 814 | 13 | 78 | No |
375 | 492 | 15 | 90 | No |
682 | 644 | 14 | 84 | Yes |
Cross-check on the illustration: its Step-I digit sums are 13, 14, 11, 15, 15, 15 - only one (14) is even, and only one Step-IV number there (84) is a multiple of 4. The parity test is consistent.
For the new input, two Step-I digit sums are even (10 and 14), so exactly two Step-IV numbers (60 and 84) are divisible by 4. Hence the answer is two.