A twin prime is a prime number that is either 2 less or 2 more than another prime number for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two.
The first few twin prime pairs are:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109) and so on
So the answer uis 71,73
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Find the remainder when 83*85*87*88*91 is divided by 41.
32^31^301
when 31 divided by 9 gives remainder 5
5 5^2 5^3 all gives the same unit digit 5
so 31^32 gives unit digit 5
same rule applicable to 31^301
when 31 divided by 9 gives remainder 4
4 4^2 4^3 4^4 =4 6 4 6 unit place repeats for every 2 times i,e for even power its uint place is 6 and for odd its 4
as 301 is odd its unit place is 4
so 31^32^301=31^4=5^4=5
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How many positive integers less than 500 can be formed using the numbers 1,2,3,and 5 for digits, each digit being used only once.
Units digits of first 4 terms are 7, 9, 3, 1. and this pattern repeats. So for every 4 terms we get one term with 3 in its unit digit. So there are total of 205/4 = 51 sets and each set contains one terms with 3 in its unit digit.
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