# Gyan Room - Number Theory - Short cuts/Questions/Concepts

This thread is reserved for sharing concepts, short cuts and good questions from Number Theory topic.

Rules:

1. Mention source (if required)
2. Provide examples for short-cuts
3. 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 9.1^8.9

another 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 factors

Example - 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)

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