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Solution: By Fermat's theorem as 24 and 7 are coprimes, remainder of 24^6 mod 7 =1. Hence,24^13 can be split up as 24^6* 24^6* 24 mod 7 = 1*1*3 mod 7 =3. Hence, remainder is 3. How is 24 mod 7 calculated here

can u plz elaborate how did they get 3 here as a remainder

24 mod 7 is 1. It is a direct result from Fermet's Theorem which says that if a and b are co-prime then a^E(b) mod b = 1.E(b) is the euler no. of b. Here 24 and 7 are coprime and euler no. for 7 is 6 so 24^6 mod 7 = 1.

Four unique even numbers are arranged in ascending order and the first two are decreased by 50%, while the last two are increased by 50%. If the original sum of the 4 numbers was 162, the new sum can be? In the solution it is mentioned that the last two numbers must be 4 greater than the first two. Why?

When arranged in ascending order, the difference between 2nd term and 3rd term must be at least 2. Now when 2nd number is reduced by 50% and 3rd term is increased by 50% , so the difference between them will increase by 2. Thus the minimum difference would be 4. Hope it clears your doubt.Since the question says can be, options are needed to tell the answer.

I am basically from commerce field,so I cant understand geometry very well....what should I do to improve geometry?

Study the basics from any book, if you are not too sure about your concepts, you can refer NCERTs of class 8th to 10th. After that practice questions regularly to get familiar with applications part. For this, you need to take mocks or practice from some other good source like previous years CAT papers.

Not able to understand what you want to ask. Could you please be more clear.

A student appears for 4 papers-english,math,physics,chemistry.Max Marks for which are in the ratio 1:1:2:2. His marks are in the ratio of4:8:13:15. If he got 80% of the total max marks, in how many papers did he got more than 80%?

Let us assume that the max marks are 50,50,100 and 100 resp.total marks are : 80% of 300 i.e. 240now divide 300 in ratio 4:8:13:15 such that the sum is 240.marks are 24:48:78:90.marks more than 80% in 02 subjects

His maximum marks would be a multiple of 6. Let it be 6x. So sum of his marks would be .8*6x = 4.8x. To simplify calculations multiply both sides by 100 so we get, max marks = 600x, His marks = 480x. Now subject wise his marks would be, 48,96,156, 180. Maximum marks in these respective subjects would be 100, 100, 200, 200. So his marks are 48/100, 96/100, 156/200, 180/200. Thus in 2 subjects he had greater than 80%.