Count them all!!
Number Theory
Level
4
For how many positive integers less than 300 is - divisible by 25?
The answer is 71.
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1 solution
n 5 − n 2 = n 2 ( n 3 − 1 )
So for 25to divide this expression either n 2 or n 3 − 1 have to be divisible by 25.
n 2 will be divisible by 25 for all n = 5 x for all integers x. But, n < 300, so x =< 11. So, there are 59 such integers.
Now, for n 3 − 1 to be divisible by 25, either (n-1) or n 2 + n + 1 . Therefore, there are 11 more integers which we get from (n-1). Now, n 2 + n + 1 cannot be a multiple of 25. This is because product of 2 numbers can never end in digit 4 or 9.
Therefore, there are (11 + 59)= 70 possible integers.
But the main thing here is that it has one more solution i.e. n=1. This is because 25 divides (1-1=0) also.
Therefore , the answer is 70 + 1= 7 1