. A positive integer n > 1 is called beautiful if n can be written in one and only one way as n = a1 + a2 + · · · + ak = a1 · a2 · · · ak for some positive integers a1, a2, . . . , ak , where k > 1 and a1 ≥ a2 ≥ · · · ≥ ak . (For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2 · 2 · 2 · 1 · 1 , so uniqueness is lost.) Find the largest beautiful number less than 100
Question
. A positive integer n > 1 is called beautiful if n can be written in one and only one way as n = a1 + a2 + · · · + ak = a1 · a2 · · · ak for some positive integers a1, a2, . . . , ak , where k > 1 and a1 ≥ a2 ≥ · · · ≥ ak . (For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2 · 2 · 2 · 1 · 1 , so uniqueness is lost.) Find the largest beautiful number less than 100
Solution
The problem is asking for the largest number less than 100 that can be expressed as both the sum and product of a unique set of numbers.
To solve this problem, we can start by checking numbers from 99 downwards and see if they meet the criteria.
However, to make the process faster, we can make some observations:
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The number must be even. This is because if the number is odd, then one of the numbers in the set must be odd. But the product of an odd number of odd numbers is odd, and the sum of any number of odd numbers is odd, so there would be at least two ways to express the number.
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The number cannot be a multiple of 4. This is because if the number is a multiple of 4, then it can be expressed as the sum and product of 2's and 1's in at least two ways.
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The number cannot be a multiple of 3. This is because if the number is a multiple of 3, then it can be expressed as the sum and product of 3's and 1's in at least two ways.
Using these observations, we can narrow down the possible numbers to check.
Starting from 99 and going downwards, the first number that meets all these criteria is 98.
However, 98 can be expressed as the sum and product of 49 and 2, as well as 49, 2, and 1. So, 98 does not meet the criteria.
Continuing downwards, the next number that meets all these criteria is 94.
94 can be expressed as the sum and product of 47 and 2. There are no other ways to express 94 as the sum and product of a set of numbers, so 94 is the largest beautiful number less than 100.
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Problem statementSend feedbackYou are given an array ‘A’ of length ‘N’. You say an index ‘i’ is beautiful if the sum of the first ‘i - 1’ elements of the array ‘A’ equals the sum of the last ‘N - i’ elements of the array ‘A’, where ‘i’ is in 1-based indexing. Now you wonder which is the leftmost beautiful index.Note: If you select the first index, then the sum of the prefix will be ‘0’, and if you select the last index, then the sum of the suffix will be ‘0’.Note: You have to print the index in 1-based indexing.For example:Let’s say the array ‘A’ = [1, 3, 1, 5], then if we select index ‘2’, the sum of the prefix is ‘1’, and the sum of the suffix is 1 + 5 = 6. Since the sum is not the same, hence index ‘2’ is not a beautiful index.Detailed explanation ( Input/output format, Notes, Images )Constraints -1 <= ‘T’ <= 101 <= ‘N’ <= 10^5 -10^9 <= A[i] <= 10^9, for 1 <= i <= ‘N’Note- Sum of ‘N’ over all test cases does not exceed 10^5.Time Limit: 1 secSample Input-1231 1 131 2 3Sample Output-12-1Explanation for Sample Input 1:For test case 1: Index ‘2’ is the leftmost beautiful index. The left sum is 1 and the right sum is also 1.For test case 2: No index is beautiful.Sample Input -2261 7 3 6 5 632 1 -1Sample Output -241
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