🔢 number theory

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 $n$.

1. The problem asks us to find numbers with rotational symmetry of order 2, meaning when rotated 180 degrees, the digits look the same. 2. We analyze digits 0-9 for rotational symm

1. The problem asks to classify the number $-1$ by choosing the correct types of numbers it belongs to. 2. First, recall the definitions:

1. The statement claims that if $n$ is a prime number, then $2n + 1$ will also be a prime number. 2. To test this, let's use an example with $n = 5$, which is a prime number.

1. The given sequence is 2, 3, 5, 7, 11, 13,... 2. This sequence consists of prime numbers: numbers greater than 1 that have no positive divisors other than 1 and themselves.

1. The problem asks for the smallest integer \(n\) such that \(2250n\) is a perfect cube. 2. First, factorize 2250 into its prime factors.

1. The problem is to list prime numbers, which are numbers greater than 1 that have no positive divisors other than 1 and themselves. 2. The first few prime numbers are:

1. The term H.C.F stands for Highest Common Factor, which is the greatest number that divides two or more numbers without leaving a remainder. 2. To find the H.C.F of given numbers

1. The problem is to convert the decimal number 14 into its binary (base 2) equivalent. 2. To convert from decimal to binary, repeatedly divide the number by 2 and record the remai