Number system
Q1. If X^7 -X^3=1234 and if X is a natural number less than 350 .Find the number of values of X satisfying this equation.
solution:
this problem is solved based on cyclic principle of power of a number.
for example natural numbers ending with 2,3,4,7,8 has cyclic behavior of its units place after every 4 intervals(n=1,2,3,4)
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
2^9=512
here we can see the units place repeats 2,4,8,6 again and again and again
exception for this logic is number ending with 5,6 and 9
for numbers ending with 5^n where n is a natural number its units place will be 5
for numbers ending with 6^n where n is a natural number its units place is 6
for numbers ending with 9^n where n is a natural number its units place can be 1or 9(cyclic pattern repeats after every 2 intervals )
we can observe X^7 -X^3 's unit place is same.
for example 2^3=8 and 2^7=128. when subtracted units place must be zero.
hence in the above question X^7 -X^3 =1234
will not have any solution because units place is not zero
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