I am gonna make account of good observations and ideas which I come across while solving problems. The proofs of below statements will not be mentioned here, Its advised to do such proofs on own for exercise. ↵
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- Lets say I have a set $S$ consisting of integers, denote its $lcm(S) = L$, I add a new element $x$ to this set $S$ , ↵
Lets deonte the new set as $S'$,where $S' = union(S , x)$ and its $lcm(S') = L'$. Can we deduce a relation between $L$ and $L'$. We can observe that either $L = L'$ or $L' >= 2*L$. ↵
- Let's say we want to find 2 numbers in an array $A[]$ with maximum common prefix bits in binary representation. Its easy to show that those two numbers always occur as adjacent numbers in $sorted(A[])$ ↵
- The number of distinct gcd prefixed/suffixed at an index in an array will never exceed $log(A_{max})$ ↵
- Let's say I have a number $X$, And I apply modulo operation as many times as I wish, i.e $X = X \% {m_i}$ for some different values of ${m_i}$. It can be shown that $X$ takes $log(X)$ distinct values until it reaches to $0$. ↵
- If $N$ times $abs()$ function appears at any problem, maybe bruteforcing all $2^N$ combinations of $+/-$ may give way to the solution sometimes.↵
- Prefix Or/And can take a maximum of $log(N)$ values.↵
- Nested totient function say $phi(phi(phi( ... (X) ... )))$ will eventually reach 1 in atmost $2log(X)$ nested functions. Useful for computing expressions like $(A^{(B^{(C^..)})})$ modulo $P$. (nested powers).↵
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- Lets say I have a set $S$ consisting of integers, denote its $lcm(S) = L$, I add a new element $x$ to this set $S$ , ↵
Lets deonte the new set as $S'$,where $S' = union(S , x)$ and its $lcm(S') = L'$. Can we deduce a relation between $L$ and $L'$. We can observe that either $L = L'$ or $L' >= 2*L$. ↵
- Let's say we want to find 2 numbers in an array $A[]$ with maximum common prefix bits in binary representation. Its easy to show that those two numbers always occur as adjacent numbers in $sorted(A[])$ ↵
- The number of distinct gcd prefixed/suffixed at an index in an array will never exceed $log(A_{max})$ ↵
- Let's say I have a number $X$, And I apply modulo operation as many times as I wish, i.e $X = X \% {m_i}$ for some different values of ${m_i}$. It can be shown that $X$ takes $log(X)$ distinct values until it reaches to $0$. ↵
- If $N$ times $abs()$ function appears at any problem, maybe bruteforcing all $2^N$ combinations of $+/-$ may give way to the solution sometimes.↵
- Prefix Or/And can take a maximum of $log(N)$ values.↵
- Nested totient function say $phi(phi(phi( ... (X) ... )))$ will eventually reach 1 in atmost $2log(X)$ nested functions. Useful for computing expressions like $(A^{(B^{(C^..)})})$ modulo $P$. (nested powers).↵
