Number Theory

Number Theory Rules

1. 3^n will always have even number of tens. For ex: 3^3={2}7 , 3^4=[8]1,3^5=[ 24]3,3^6=[72]9 2. A sum of 5 consecutive whole numbers will always be divisible by 5. 3. The diff b/w two numbers Xy - yx will be divisible by 9. 4. The square of an odd number when divided by 8 will always leave a remainder of 1 5. The product of three consecutive natural numbers is divisible by 6. 6. The product of 3 consecutive natural nos the fist of which is even is divisible by 24. 7. Odd*Odd=Odd Odd*Even=Even Even*Even=Even 8. All nos not divisible by 3 have the property that their square will have an remainder of 1 when divided by 3. 9. (a^2+b^2) /(b^2+c^2) = a^2/b^2 if a/b = b/c 10. The product of any r consecutive integers is divisible by r!. 11. If m& n are two integers then (m+n)! is divisible by m!n!. 12. Any no. written in the form 1o^n-1 is divisible by 3& 9. 13. (a)^n /(a+1) leaves a remainder of a if n is odd 1 if n is even (a+1)^n/a will always give a remainder of 1 14. For any natural number n. n^5 has the same unit digits as n have. 15. For any natural number, n^3-n is divisible by 6.