🔢 number theory

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

1. **Problem Statement:** Find the digit $D$ such that the last digit of $(54D)^{100}$ is 1. 2. **Understanding the problem:** The last digit of a number raised to a power depends