To directly find the gcd of two numbers, it is necessary to represent both numbers as a product of prime powers. Only then, one can match the powers and primes. Additionally, since the two numbers are coprime, they do not have any prime factors in common. Question: Determine the highest common factor between $2^4cdot3^4cdot25cdot7$ and […]

# Category: Elementary number theory

## Converse of Euler’s Totient Theorem: A Proof

Computing the modular inverse using the method of factorization of m is not practical for large numbers. Another solution is to prove that φ(m) ≡ 1 (mod m) using the set of numbers A, which consists of φ(m) numbers that are relatively prime to m and are represented as {$n_1, n_2, … n_{phi(m)}$}$pmod{m}$. Solution 1: […]

## Calculating the value of $29^{25}$ modulo 11

Solution 2 can be done numerically in pari-gp, or theoretically using multiplicative order, discrete logarithm, and the Chinese remainder theorem. In Solution 3, we are asked to find what value times -2 is equal to -(-1)=1, which is the same as finding what times 2 is equal to 12 (given that 9=-2 mod 11). The […]

## Zero: Is it a Part of the Set of Natural Numbers?

Generally speaking, 0 is not considered a natural number and is excluded when talking about positive/negative numbers People commonly start counting from 1, 2, 3, etc. and exclude 0 The 1st number is 1, not 0 Solution 1: While a natural number does not have a leading or ending digit, when represented in a (positional) […]

## A formula for computing exponential sums

The expression $s(s^L)’$$ can be rewritten as: $$sum_{L=0}^M L^2s^L=0^2+1^2 s + s^2 left(sum_{L=2}^{M} s^L right)”+s left(sum_{L=2}^{M} s^L right)’$$ Another method is to manipulate the inner expression to resemble a derivative: $$ begin{align} sum_{L=0}^Mleft(Ls^{L-1}right)sL & =ssum_{L=0}^Mleft(partial_ss^Lright)L\ & =spartial_ssum_{L=0}^Ms^LL\ & =spartial_ssum_{L=0}^Mleft(Ls^{L-1}right)s\ & =spartial_sleft(ssum_{L=0}^Mleft(Ls^{L-1}right)right)\ & =spartial_sleft(ssum_{L=0}^Mpartial_ss^Lright)\ & =spartial_sleft(spartial_ssum_{L=0}^Ms^Lright)\ & =spartial_sleft(spartial_sfrac{s^{M+1}-1}{s-1}right)\ end{align}$$ After simplifying the above expression, apply the […]

## What is the antonym of a prime number?

The number of checks required to verify the prime factors of $N-1$ is at most $O(log N)$, with each check being a modular exponentiation in $mathbb{Z}/Nmathbb{Z}$ taking $O(log^2 N)$ time. Therefore, the total complexity is $O(log^3 N)$. In $mathbb{Z}$, a prime is any non-zero, non-invertible element $p$ of the ring such that for all $a, […]

## Are Natural Numbers and Whole Numbers the Same?

The terms integers, whole numbers, and natural numbers may be different, but they are related in some way. Integers refer to all numbers on the number line, including positive, negative, and zero. Whole numbers only include positive numbers and zero, while natural numbers only include positive numbers. If zero is removed from whole numbers, they […]

## Proving Divisibility of Numbers: Is it Possible to Determine if a Number is Divisible by 6?

The number of ways to fill in the other digits (from left to right) varies depending on the last digit of the number. If the last digit is 0, there are 24 possible ways. If the last digit is 2 or 4, there are 18 possible ways to fill in the other digits. Question: Demonstrate […]

## Total of the initial 100 natural numbers

Initially, breaking mathematical equality is required by claiming that a divergent series can be equal to a particular value. Although this is an enjoyable way to manipulate divergent series, it does not adhere to the rigorous principles of mathematical equality. Solution 1: Do you need to calculate the sum of $a$ plus all consecutive integers […]

## Is -1(mod 13) equivalent to 12(mod 13)?

In $mathbb{Z}^*_{13}$, the group law is “multiplication modulo 13”, which includes all invertible elements modulo 13. The group law fulfills certain properties, such as being associative and having a neutral element and opposite element for each element in the group. The neutral element in this case is 0, and for example, 4 has an opposite […]