🔢 number theory

1. **State the problem:** We need to find the greatest common divisor (gcd) $d$ of $a=135$ and $b=59$, and express $d$ as a linear combination $d = sa + tb$ where $s > 0$ is as sma

1. The problem asks us to identify which number among 4, 5, 9, and 15 is composite. 2. A composite number is a positive integer greater than 1 that has more than two distinct posit

1. **Problem 36:** Determine which of the given numbers divides $2^{15}$. 2. Recall that $2^{15} = 32768$ and is a power of 2, so its prime factorization is only 2's.

1. The problem asks us to classify the numbers 1, 6, 25, 29, 31, 33, 49, 51, and 53 into three categories: Prime, Composite, and Neither. 2. Definitions:

1. The problem asks for the prime number of 4. 2. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

1. **State the problem:** We need to classify the numbers in set A = { -6, \frac{1}{2}, -1.333... (3's repeat), \pi, 2, 5 } into (a) Natural numbers, (b) Integers, (c) Rational num

1. **Stating the problem:** We want to analyze the expression $7^n - 1$ where $n \in \mathbb{Z}^+$ (positive integers) and determine its divisibility by 6. 2. **Formula and rules:*

1. **Problem statement:** Find all positive integers $k>1$ such that there exists a positive integer $n$ with the property that $\binom{n}{k}$ is divisible by $n$, but for all $m$

1. **State the problem:** We are given the number 67 and need to understand or work with it as per the user's request. 2. **Since the user only provided the number 67 without a spe

1. **Problem Statement:** Show that the sum of a rational number and an irrational number is irrational. 2. **Definitions:**

1. The problem is to convert an integer from one base to another. 2. The general method involves two main steps: first, convert the number from the original base to base 10 (decima

1. **Problem statement:** Given a prime $p \in \mathbb{F}$ and the function

1. **Determine whether the number 57 is prime or composite.** A prime number has exactly two distinct positive divisors: 1 and itself.

1. The problem is to understand what an odd number is and to provide examples of odd numbers up to one hundred million. 2. An odd number is an integer which is not divisible by 2.

1. The problem is to understand what an even number is and to see examples of even numbers up to one hundred million. 2. An even number is any integer that can be exactly divided b

1. The problem asks us to identify which numbers among 15, 13, 9, and 8 are prime. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 a

1. The problem is to identify which numbers among 49, 47, 13, and 21 are prime. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and

1. **State the problem:** Solve the congruence equation $$13x \equiv 71 \pmod{380}$$. 2. **Formula and rules:** To solve $$ax \equiv b \pmod{m}$$, we need to find the modular inver

1. **State the problem:** Find the greatest number that divides 43, 91, and 183 leaving the same remainder in each case. 2. **Key idea:** If a number $d$ divides these numbers leav

1. **State the problem:** We need to show that every term of the sequence defined by $U_k = 2(4^k) + 1$ is divisible by 3 for all integers $k \geq 0$. 2. **Recall the divisibility

1. সমস্যাটি হলো: ১০০৮ এর মোট ভাজক (divisors) কতটি আছে তা নির্ণয় করা। 2. ভাজক নির্ণয়ের সূত্র: একটি সংখ্যার মৌলিক গুণনীয়করণ করলে যদি $$n = p_1^{a_1} \times p_2^{a_2} \times \cdots