If m and n are odd positive integers, then m2 + n2 is even, but not divisible by 4. Justify.

Asked by Last Modified  

5 Answers

Learn Mathematics +1 Class IX-X Tuition

Follow 5
Answer

Please enter your answer

Gurukul Education Center

Since m is odd positive integer then we can write m=2a+1 and n=2b+1, such that a,b >0 m2+n2=(2a+1)^2+(2b+1)^2 = 4a^2+1+4a+4b^2+1+4b = 4a^2+4b^2+4a+4b+2 = 2(2a^2+2b^2+2a+2b+1) Above eqn is divisible by 2 but not divisible by 4.
Comments
Dislike Bookmark

Tutor

Let m=2a+1, n=2b+1 be two odd positive integers Now m^2+n^2=(2a+1)^2+(2b+1)^2 =4a^2+1+4a+4b^2+1+4b =4a^2+4b^2+4a+4b+2 =2(2a^2+2b^2+2a+2b+1) Therefore m^2+n^2 is divisible by 2 but not with 4 Example let m=1, n=5 m^2+n^2=(1×1)+(5×5)=1+25=26 26 is divisible by 2 but not with 4
Comments
Dislike Bookmark

Maths Tutor

Let the two odd positive numbers be x = 2k + 1 a nd y = 2p + 1 Hence x2 + y2 = (2k + 1)2 + (2p + 1)2 = 4k2 + 4k + 1 + 4p2 + 4p + 1 = 4k2 + 4p2 + 4k + 4p + 2 = 4(k2 + p2 + k + p) + 2 Clearly notice that the sum of square is even the number...
read more
Let the two odd positive numbers be x = 2k + 1 a nd y = 2p + 1 Hence x2 + y2 = (2k + 1)2 + (2p + 1)2 = 4k2 + 4k + 1 + 4p2 + 4p + 1 = 4k2 + 4p2 + 4k + 4p + 2 = 4(k2 + p2 + k + p) + 2 Clearly notice that the sum of square is even the number is not divisible by 4 Hence if x and y are odd positive integers, then x2 + y2 is even but not divisible by 4 cross check let m= 3 and n=5 3^2+5^2=34 34 is not divisible by 4 hence prove read less
Comments
Dislike Bookmark

Since m is odd m*m is also odd. This imply m*m+n*n is of the form 4*k+2 (Since m*m is of the form 4*k+1).
Comments
Dislike Bookmark

Tutor

M=1,n=3 then m2+n2=one2+three2=10,10 is not divisible by 4
Comments
Dislike Bookmark

View 3 more Answers

Related Questions

Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers?
Let numbers are 5x and 6x (5x-8)/ (6x-8) = (4/5) => 5(5x-8)= 4(6x-8) => 25x-40 = 24x-32 => 25x-24x= 40-32 => x= 8 Two numbers are 5x 8= 40 and 6x8 = 48
Sensoph
If sin A=1/2, then find the value of cos A.
sin (A/2)= 1/2 =sin (A/2)=sin (30°)=sin(180-30) A/2= 30° or 150° A= 60° or 300° sin(A)=sin (60°)=√3/2 sin(A)=sin (300°)=sin(360°+ -60°)=sin(-60°)=-sin (60°)=(-√3)/2 sin...
Malvika
Why is algebra taught in high school?
To learn basics
Darshan
0 0
7
A man buys a cycle for Rs 1400 and sells it at a loss of 15%. What is the selling price of the cycle?
Loss = Loss%/100 * CP = (15/100) * 1400 = 210 SP = CP - Loss = 1400 - 210 = 1190
Shubham
How many edges are there in a cylinder?
2 edges.
Thileshwari
0 0
6

Now ask question in any of the 1000+ Categories, and get Answers from Tutors and Trainers on UrbanPro.com

Ask a Question

Related Lessons

Class-10 L-5 Arithmetic Progressions Ex-5.3 Q11

Ashish Bhatia

0 0
0
Class-10 L-4 Quadratic Equations Ex-4.3 Q3

Ashish Bhatia

0 0
0
Class-10 L-5 Arithmetic Progressions Ex-5.1 Q1 (i)

Ashish Bhatia

0 0
0
Class-10 L-5 Arithmetic Progressions Ex-5.1 Q2 (iii) (iv)

Ashish Bhatia

0 0
0
Class-10 L-5 Arithmetic Progressions Ex-5.1 Q4 (xiii) (xv)

Ashish Bhatia

0 0
0

Recommended Articles

Meet Raghunandan.G.H, a B. Tech Tutor from...

Raghunandan is a passionate teacher with a decade of teaching experience. Being a skilled trainer with extensive knowledge, he provides high-quality BTech, Class 10 and Class 12 tuition classes. His methods of teaching with real-time examples makes difficult topics simple to understand. He explains every concept in-detail...

Read full article >

Meet Swati, a Hindi Tutor from Bangalore

Swati is a renowned Hindi tutor with 7 years of experience in teaching. She conducts classes for various students ranging from class 6- class 12 and also BA students. Having pursued her education at Madras University where she did her Masters in Hindi, Swati knows her way around students. She believes that each student...

Read full article >

Meet Sandhya R, a B.Sc tutor from Bangalore

Sandhya is a proactive educationalist. She conducts classes for CBSE, PUC, ICSE, I.B. and IGCSE. Having a 6-year experience in teaching, she connects with her students and provides tutoring as per their understanding. She mentors her students personally and strives them to achieve their goals with ease. Being an enthusiastic...

Read full article >

Quest Academy - Institute of the month

Quest Academy is a professional Bangalore based NEET and JEE (Main + Advanced) training institute. The academy was incorporated in 2015 to cater to the needs of students, who aim to crack competitive exams by connecting with the best brains around. The institute helps students enhance their skills and capabilities through...

Read full article >

Find Grade 10 Tuition near you
Grade 10 Tuition in Delhi
Grade 10 Tuition in Bangalore
Grade 10 Tuition in Hyderabad
Grade 10 Tuition in Kolkata
Grade 10 Tuition in Chennai
Grade 10 Tuition in Mumbai
Grade 10 Tuition in Pune
Grade 10 Tuition in Noida
Grade 10 Tuition in Lucknow
Grade 10 Tuition in Gurgaon
Online Tuition Class 10

Looking for Class 10 Tuition ?

Learn from the Best Tutors on UrbanPro

Are you a Tutor or Training Institute?

Join UrbanPro Today to find students near you