🔢 number theory

1. **Problem Statement:** Prove that among any 16 consecutive integers, there is at least one integer that is relatively prime (shares no common divisor greater than 1) to each of

1. Mari kita nyatakan masalahnya: kita ingin mencari solusi bilangan bulat positif dan bilangan bulat tak negatif dari persamaan Diophantine $$x + y + z = n$$ dengan $n = 10$. 2. U

1. Masalah yang diberikan adalah mencari solusi bilangan bulat positif dari persamaan Diophantine $$x + y + z = n$$ dengan $$n = 10$$. 2. Kita tahu dari contoh bahwa solusi adalah

1. The problem asks for the digit in the 150th position when writing natural numbers in sequence: 123456789101112... 2. We write numbers consecutively and count digits: single-digi

1. نبدأ بفهم المعادلة المعطاة: $a \equiv 3b \pmod{10}$ تعني أن الفرق بين $a$ و $3b$ يقبل القسمة على 10. 2. المعادلة $<1+3i> = 10z$ تعني أن العدد العقدي $1+3i$ مضروب في عدد صحيح $z$

1. المشكلة: لدينا العلاقة $a \equiv 3b \pmod{10}$ ونريد فهم ما إذا كان هذا يعني أن $a = 10k + 3b$ حيث $k$ عدد صحيح. 2. القاعدة: تعني $a \equiv 3b \pmod{10}$ أن الفرق $a - 3b$ يقبل

1. The problem is to understand what prime numbers are. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

1. **State the problem:** Find the least number $N$ such that: - When divided by 52, remainder is 33.

1. **State the problem:** Solve the system of congruences: $$x + 5y \equiv 3 \pmod{9}$$

1. **Restate step 5:** We substitute the expression for $y$ from step 4 into the second congruence. 2. From step 4, we have:

1. **Stating the problem:** Solve the system of congruences: $$2x + y \equiv 1 \pmod{6}$$

1. The problem is to understand the number 13 and explain its properties. 2. 13 is a natural number that comes after 12 and before 14.

1. Тодорхойлъё: $x \equiv 2 \pmod{5}$ гэдэг нь $x$-ийг 5-т хуваахад үлдэгдэл 2 гарна гэсэн үг юм. 2. Энэ тэгшитгэлийг өөрөөр бичихэд, $x$ нь 5-ийн ямар нэгэн бүхэл үржвэр дээр 2 нэ

1. Тодорхойлолт: Бид хамгийн бага натурал тоог олох ёстой, тэр тоог 5-д хуваахад үлдэгдэл 2, 7-д хуваахад үлдэгдэл 3, 9-д хуваахад үлдэгдэл 4 байна. 2. Математик бичиглэл: Хэрвээ $

1. مسئله: تعداد اعداد اول کوچکتر از 100 را پیدا کنید. 2. تعریف عدد اول: عدد اول عددی طبیعی بزرگتر از 1 است که فقط بر 1 و خودش بخش‌پذیر باشد.

1. **Problem Statement:** Find the digital root of a number by repeatedly summing its digits until a single digit (1 to 9) remains. 2. **Definition:** The digital root of a number

1. **Problem statement:** Find the set of solutions for the linear congruences: i. $x \equiv 3 \pmod{5}$

1. **State the problem:** Show that $11^9 + 9^{11}$ is divisible by 10. 2. **Recall the divisibility rule:** A number is divisible by 10 if its last digit is 0, which means the num

1. **Problem statement:** A number when divided by 30 leaves a remainder of 8. We need to find what number should be added to this number to make it a multiple of 6. 2. **Understan

1. **Stating the problem:** We need to identify which numbers among 153, 257, 408, and 441 are definitely not perfect squares without performing detailed calculations. 2. **Importa

1. **Problem Statement:** Find the possible one's digits of the square roots of the given numbers: 9801, 059856, 998001, 657666025. 2. **Important Rule:** The one's digit of a perf