NUMBERS - Important Formulae/Notes

(1) Product of any 3 consecutive natural numbers is divisible by 6
Eg: 4*5*6=300 (div by 6)

(2) Any prime no. That is greater than 3 can be expressed in the following form
N = (6k + 1) or (6k – 1)

(3) Find the highest power of “a” that divides N!

Now the highest power of "a" that divides N! = (x+y+z)

(4) If a number “N” is divided by a,b,c and leaves reminders a,b,c, then,
N = LCM(a,b,c) + A
A = (bz+y)a + x

(5) A perfect number is a number that can be expressed in the below format,
where p = any prime no

(6) If
(a-1)n = ka + 1 → (a is even)
(a-1)n = ka - 1 → (a is odd)
(a+1)n = ka + 1 → (a is even or odd)

(7) Sample Q: Find the greatest power of 30 in 200!
Solution: First divide 30 its atomic factors 30 = 5*3*2
Now greatest power of 30 in 200! = greatest power of 5 in 200!
To find the greatest power of 5 in 200!,

Now ans = Greatest power of 30 in 200! = 40+8+1 = 49