Fun with Mathematics..!

Finding Digit? Finding Last Digit(Units Digit) of X^n , Where x is an integer and n is a positive integer Example - 12*12 = LD(4)                       2^3 = 8                       LD(12^3) = __8 The last digit of X^n depends only on the last digit of x itself raised to the power. Last Digits of the powers of any digit follow a cyclic pattern i.e. they repeat after a certain number of steps. If the last digit of the base x is 0 or 1 or 5 or 6 then the last digit of the result when x is raised to a positive power integer power is 0,1,5 and 6 respectively. If the last of the base x is 4 or 9 then                  For 4, every odd power will give the last digit of 4 and every even power will give last digit 6                F or 9, every odd power will give the last d...

DIVISOR and REMAINDERS Rem <= Divisor Rem(N/D) = r , then N= Dx + r remainders can be -ve also can be converted into +ve by adding divisor LCM and HCF of two natural numbers - x,y HCF = h then numbers are x= ha, y = hb a,b Co-prime LCM - hab xy = lcm*hcf LCM and HCF for a fraction -  LCM of fractions = LCM of all Numerators/ HCF of all Denominators HCF of fractions = HCF of all Numerators / LCM of all Denominators (fraction is in most reduced format) LCM and HCF via prime factorization = For LCM, take all the prime bases and raise them each to their highest powers For HCF, take prime bases that are common to all numbers and raise them each to their lowest powers LCM - Type 1 ( Common Remainders) Divisor 👍 Remainder 👍 (Same remainder for all divisor) Number 👎 Sol -  LCM(Divisors) + Remainder   👉 Least Number(N) LCM Type 2 ( Common Difference between divisors and their respective remainder) Divisor 👍 Remainder 👍 Number 👎 Sol - Number(N) = (k*LCM(Divisors)) - ...