🔢 number theory

1. The problem is to find the link between two prime numbers. 2. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

1. **State the problem:** We need to find the number of six-digit palindromic integers divisible by 15. 2. **Understand palindromes:** A six-digit palindrome has the form $ABC CBA$

1. **State the problem:** We need to determine which classifications apply to the number $-9$ from the options: rational number, real number, whole number, and integer. 2. **Recall

1. Problem: Ana starts with 8 and counts by 5s. Her first three numbers are 8, 13, and 18. Josh starts with a whole number other than 8 and counts by a whole number other than 5. S

1. The problem is to verify the property of the greatest common divisor (GCD) for three integers $a$, $b$, and $c$: $$\text{ggT}(a,b,c) = \text{ggT}(a, \text{ggT}(b,c))$$ 2. The GC

1. **Problem statement:** Find the greatest integer $q$ such that $4^q$ divides the product $p = 1 \times 2 \times 3 \times \cdots \times 1000$, which is $1000!$. 2. **Understandin

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

1. The problem asks to convert the decimal number 314 (base 10) to base 6. 2. A base (or radix) is the number of unique digits, including zero, used to represent numbers in a posit

1. **State the problem:** We need to find the digit $x$ in the base 5 number $34x1_5$ such that the number is divisible by 31 in decimal. 2. **Convert the base 5 number to decimal:

1. **State the problem:** Find the number of distinct positive divisors of $30^4$ excluding 1 and $30^4$ itself. 2. **Prime factorization:** First, express 30 as a product of prime

1. Statement of the problem. We seek all positive integers $x,y,z$ such that $x+y+z$ is prime and $xy+yz+zx$ divides $x^2+y^2+z^2$.

1. **State the problem:** Jason bought a sack of rice weighing more than 25kg but less than 60kg. He packed the rice into 5kg bags and had 1kg left over.

1. The problem asks to identify properties of irrational and rational numbers. 2. An irrational number is defined as a number that cannot be expressed as a ratio of two integers $\

1. The problem is to explain the number 11 in base 10 (decimal) and how it can be understood or represented. 2. In base 10, the number 11 means 1 ten and 1 one, which can be writte

1. **State the problem:** We need to find two prime numbers that add up to 30.

1. **Stating the problem:** We have three numbers with HCF (Highest Common Factor) 8 and LCM (Least Common Multiple) 2520.

1. **Problem statement:** Prove that for any positive integer $n$, the fraction $\frac{21n + 4}{14n + 3}$ is irreducible, meaning it cannot be simplified further. 2. **Key idea:**

1. **State the problem:** Find all positive integers $n$ such that $n^3 + 2n + 1$ is a perfect square. 2. **Set up the equation:** Let $k^2 = n^3 + 2n + 1$ where $k$ is an integer.

1. **Problem statement:** Prove that for any positive integers $m$ and $n$, the product $$(36m + n)(36n + m)$$ can never be a power of 2. 2. **Recall the definition:** A power of 2

1. Problem: Find the remainder when $7^{5284}$ is divided by 5. 2. Formula: Use modular arithmetic and Euler's theorem or Fermat's little theorem.

1. The problem is to create an equation using the digits 1 through 9 exactly once each, arranged to produce the highest possible integer value. 2. To maximize the integer value, we