Gyan Room  Number Theory  Short cuts/Questions/Concepts

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This thread is reserved for sharing concepts, short cuts and good questions from Number Theory topic.Rules:
 Mention source (if required)
 Provide examples for shortcuts
 Posts not related to Number Theory will be removed.

Concept: f(x) = x^1/x is maximum at x = e ( e = 2.71) and f(x) will decrease if you go either side of x = e
We can use the same trick to compare a^b and b^a
Closer the base (a or b) is to e, higher the value will be.example:
9.1 ^ 8.9 and 8.9 ^ 9.1
both these values are greater than e, and 8.9 is closer to e than 9.1, so 8.9^9.1 is higher than 9.1^8.9another classic one,
which is greater, e^PI or PI^e ?
we know e = 2.7 and PI = 3.14 => e^PI is greater.This trick is especially handy when all the given numbers belongs to the same side of e, where we can do the comparison in no time.

Number of ordered pairs possible for LCM = N = P1^a x P2^b x P3^c
= [ (a+1)^2  a^2] [b+1)^2  b^2] [ (c+1)^2  c^2]
= (2a + 1 ) ( 2b + 1) (2c + 1)
Where, a, b and c are power of prime factorsExample  If LCM of two numbers is 360, how many such ordered pairs are possible?
360 = 2^3 * 3^2 * 5
so our formula is (2a + 1)(2b + 1)(2c + 1)
where a , b and c are power of prime
So [(2 * 3 + 1)(2 * 2 + 1)(2 * 1 + 1)
7 * 5 * 3 = 105

To find LCM and HCF of (a/b) and (c/d) the generalized formula will be:
H.C.F = H.C.F of numerators / L.C.M of denominators
L.C.M = L.C.M of numerators / H.C.F of denominators

a x b = HCF (a, b) x LCM (a, b)
If a and b are co primes then HCF (a, b) = 1
So for co prime numbers, a x b = LCM (a, b)