🔢 number theory

1. The problem asks to find the smallest prime number $p$ such that the number $p^2 + p + 1$ is not prime. 2. We start testing prime numbers in increasing order:

1. **State the problem:** We want to prove that for all positive integers $n$, the product $n(n+1)(n+2)$ is divisible by 6. 2. **Understand divisibility by 6:** A number is divisib

1. We are given the equation $263 + 441 = 714$ and asked to determine the base in which this equation is true. 2. Let's represent the digits in base $b$ and convert them to decimal

1. Stating the problem: We want to find the smallest number by which 243 should be multiplied so that the product is a perfect cube. 2. Prime factorize 243:

1. We are asked to find the number of pairs of positive integers \(n\) and \(m\) such that \(1! + 2! + 3! + \cdots + n! = m^2\). 2. Let \(S_n = 1! + 2! + 3! + \cdots + n!\).

1. Problem: Explore rules for sums, products, and differences involving even and odd numbers. 2. Rule (i): Sum of two even numbers.

1. **State the problem:** We need to find all pairs of positive integers $(n,m)$ such that the sum of factorials from $1!$ to $n!$ equals a perfect square $m^2$, that is, $$1! + 2!

1. Nyatakan masalah: Cari nilai $x$ jika Faktor Sepunya Terbesar (FSTB) bagi 4, 8, dan $x$ ialah 2. 2. Ingat bahawa FSTB tiga nombor ialah faktor terbesar yang sama bagi ketiga-tig

1. **State the problem:** We need to find the number of pairs of positive integers $(n,m)$ such that the sum of factorials from $1!$ to $n!$ is a perfect square, i.e., $$1! + 2! +

1. The problem asks to find which number divides the expression $$1^n + 2^n + 3^n + 4^n$$ given that $$n$$ is not divisible by 4. 2. We analyze the behavior of $$1^n + 2^n + 3^n +

1. **Problem statement:** Given an integer $n$ which is not divisible by 4, determine which number divides the sum $$1^n + 2^n + 3^n + 4^n$$. 2. To solve this, let's analyze the ex

1. **Stating the problem:** We want to find the largest number of chicken nuggets that cannot be bought using any combination of packs of 6 and 13. 2. **Explanation:** This is a cl

1. The problem asks us to find the number of pairs of positive integers $n$ and $m$ such that $$1! + 2! + 3! + \cdots + n! = m^2.$$\n\n2. We analyze the left-hand side sum for smal

1. **State the problem:** Find the largest prime factor of 999936. 2. **Start with factorization:** Recognize that 999936 is close to 1000000, which is $10^6 = 2^6 \times 5^6$. Let

1. The problem states: For an integer $n$ not divisible by 4, find the divisor of the sum $$1^n + 2^n + 3^n + 4^n.$$ 2. We will analyze the sum modulo several divisors to see which

1. **Problem statement:** We want to find the largest number of chicken nuggets that cannot be purchased using packs of 6 or 13 nuggets. 2. This is a classic example of the Frobeni

1. **State the problem:** We need to find the greatest possible cost for a single ticket given two total prices for groups of tickets: ₱975 and ₱1170. 2. **Understand the problem:*

1. **State the problem:** We want to find the greatest number of cards per page such that the 54 hockey cards, 72 baseball cards, and 63 basketball cards can each be evenly divided

1. Determine whether 21, 25, and 33 are pairwise relatively prime. Step 1: List prime factors.

1. We are asked to find the last two digits of $7^{5^6}$. 2. The last two digits of a number correspond to the number modulo 100, so we want to find $7^{5^6} \bmod 100$.

1. **Stating the problem:** Given $n$ integers $a_1, a_2, \dots, a_n$, we need to prove there exists a nonempty subset of $\{a_1, a_2, \dots, a_n\}$ whose sum is divisible by $n$.