## 16 AprThe Number of Solutions of a System of Linear Equations

The standard form of a linear equation in $$n$$ unknowns $$x_1,x_2,\dots ,x_n$$ is $$a_1x_1+a_2x_2+\dots +a_nx_n=b,$$ where $$a_1,a_2,\dots ,a_n$$ and $$b$$ are constants. Here constants mean some real numbers (these constants may come from any number field). A collection of one or more linear equations of same variables is called...

## 06 AprIf p is a prime then √p is irrational

To prove √p is irrational, where p is a prime, we will need the following theorem: Theorem: If $$p$$ is prime then $$p|ab,$$ then $$p|a$$ or $$p|b.$$ Proof: If $$p|a,$$ we are done. So let us assume that $$pmid a.$$ Therefore, $$gcd(p,a)=1.$$ Hence, by Euclid’s lemma, $$p|b.$$...

## 07 FebINMO 2013

The 28th Indian National Mathematical Olympiad (INMO) was held on 3rd February, 2013 across the nation. The questions asked in the exam are provided below. We shall publish solutions to the problems shortly. INMO 2013 [ad#ad-2]...

## 01 Febপেলিনড্ৰম সংখ্যা (Palindrome Number)

যিবোৰ শব্দ, বাক্য বা ছন্দ সচৰাচৰ পঢ়াৰ ধৰণতকৈ বিপৰীত ফালৰ পৰা, অৰ্থাৎ সোঁফালৰ পৰা বাওঁফাললৈ পঢ়িলে একেই হয় সেইবোৰক পেলিনড্ৰম বোলে। যেনে- Refer, level, নৱজীৱন। তথ্য অনুসৰি, মধ্যযুগত ইউৰোপ, আমেৰিকাত পেলিনড্ৰম আৱিস্কাৰ কৰাতো এটি খেলৰ দৰে আছিল আৰু গ্ৰীক, লেটিন আৰু ৰোমান ভাষাত ইয়াৰ চৰ্চা...

## 28 JanFind the Remainder

1) The remainder when $$3^{31}$$ is divided by 5 is A) 2 B) 3 C) 4 D) 1 (Hint: - gcd(3,5)=1 and $$\phi (5)=4,$$ thus $$3^4\equiv 1(mod 5).$$) 2) The reminder when $$5^{99}$$ is divided by 13 is A) 0 B) 8 C) 1 D) 4   3) The reminder when...

## 28 JanNumeral System

1) The symbol $$25_b$$ represents a two-digit number in base b. If the number $$52_b$$ is double the number $$25_b,$$ then b is: [Asked in “FMS”.] A) 7 B) 8 C) 9 D) 11   2) Convert the decimal number 22 to binary number.   3) Convert the decimal number...

## 23 JanH.C.F. and L.C.M.

1) The H.C.F. of $$\frac{9}{4},\frac{3}{20},\frac{9}{10} and \frac{15}{16}$$ is: A) $$\frac{3}{80}$$ B) $$\frac{3}{2}$$ C) $$\frac{1}{40}$$ D) $$\frac{45}{2}$$   2) The L.C.M. of $$\frac{2}{3},\frac{3}{20},\frac{9}{10} and \frac{15}{16}$$ is: A) $$\frac{3}{80}$$ B) $$\frac{3}{2}$$ C) $$\frac{1}{15}$$ D) $$\frac{45}{2}$$   3) The H.C.F. of 1.26, 1.155, 0.21 and 0.315 is: A) 0.05 B) 0.105 C) 1.05 D) 0.03...

## 23 JanProbability

1) $$x_{1}, x_{2}, x_{3},\dots x_{50}$$ are fifty real numbers such that $$x_{r}< x_{r+1}$$ for r = 1, 2, 3...

## 23 Janসাধাৰণ গণিত (১)

প্ৰশ্নসমূহ:- ১) ১.৪২৮৫৭১৪২৮৫৭১৪২৮৫৭১...

## 23 JanCoding and Artificial language (Aptitude test)

1) In a certain code ‘1 7 2 9’ means ‘in a programming language’, ‘7 9 3’ means ‘as a set’ and ‘2 3 4 5 6 7’ means ‘in the set as an element’. What is the code for ‘a’? A) 1 B) 7 C) 9 D) 3   2)...