341Its easy to see that a4≡1 mod 16, mod 5 and mod 3 if a is prime to 2,5 and 3.
Since 3,5,16 have no common divisor greater than 1, a4≡1 mod 240
Let a and b be two primes having at least two digits, such that a > b.
Show that 240 divides a4-b4.
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5 Answers
11Any prime's 4th power is congruent to 1 (mod240).....Any prime meaning primes >3,5,7....
62Well what soumik has written is true.. but i guess you would also want the proof for the same!
Lets try and do that...
First of all, any prime number can be written as 60K±1/ 60k±7/ 60k±11/ 60k±13/ 60k±17/ 60k±19/ 60k±23/ 60k±29
Thus, p2 is of the form 120k+1/ 120k+49/ 120k+1/ 120k+49/ 120k+49/ 120k+1/ 120k+49/ 120k+1
which is either 120k+1 or 120k+49
Thus, p4 is of the form 240k+1 or 240k+1
Now follow from soumik's proof...
3411thanks for the proof..
now how to prove that 240 is the least positive integer with that property?