🔢 number theory

1. **Problem statement:** Find the least number which when divided by 24 and 32 leaves a remainder of 6 in each case. 2. **Understanding the problem:** If a number $N$ leaves a rem

1. Problem: Find all solutions for the system of linear congruences using the Chinese Remainder Theorem (CRT) or substitution method as specified. 2. For part (a):

1. **State the problem:** Solve the system of congruences: $$x \equiv 1 \pmod{3}, \quad x \equiv 2 \pmod{5}, \quad x \equiv 3 \pmod{7}$$

1. Мәселені түсіндіру: Ең кіші ортақ еселігі (ЕКОЕ) 15 болатын натурал сандар жұптарының санын табу керек. 2. ЕКОЕ анықтамасы: Екі санның ең кіші ортақ еселігі — бұл екі санның еке

1. **State the problem:** Solve the system of congruences: $$3x + 13y \equiv 8 \pmod{55}$$

1. **Problem Statement:** We are given a positive integer $n$ and three statements about divisibility involving $n$ and $n^2$. We need to determine which statements are true. 2. **

1. समस्या: वह सबसे छोटी संख्या $x$ ज्ञात करें जो 12, 16, 18, 20 और 25 से विभाजित करने पर प्रत्येक बार शेष 4 देती है, और $x$ 7 से विभाजित होती है। 2. सूत्र और नियम: यदि कोई संख्या $

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 $