prove that if x,y,z are positive integers such that (x,y)=1 and x^2+y^2=z^4

then xy is divisible by 7

show that the condition (x,y)=1 is necessary/

(x,y) means their gcd.

## Search found 41 matches

- Mon Mar 07, 2011 8:27 am
- Forum: Number Theory
- Topic: divisibility
- Replies:
**1** - Views:
**897**

- Sat Feb 12, 2011 12:06 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2011 National Olympiad: Problemsets
- Replies:
**7** - Views:
**7579**

### Re: BdMO 2011 National Olympiad: Problemsets

i have solved one :D its number 4. i have proved (1006)^2011 is greater.(just counting power) number 1 is rather easy.i did it by induct on n.but i am not sure wheather it is correct. however n=1 the statement can be varified as follow 1+4=2+3 suppose for some m we can fulfil the statement for 4m nu...

- Thu Feb 03, 2011 3:34 pm
- Forum: Higher Secondary Level
- Topic: find the ratio
- Replies:
**0** - Views:
**799**

### find the ratio

triangle ABC has the property that all of its angles are equal to 60degre.

D is on side BC such that CD=2BD.

if CH is perpendicular to AD then whats the ratio of

<DBH/<DAB....?

D is on side BC such that CD=2BD.

if CH is perpendicular to AD then whats the ratio of

<DBH/<DAB....?

- Tue Feb 01, 2011 1:43 pm
- Forum: H. Secondary: Solved
- Topic: Dhaka Higher Secondary 2010/2 (Secondary 2010/9)
- Replies:
**17** - Views:
**7284**

### Re: Dhaka Higher Secondary 2010/2 (Secondary 2010/9)

yes moon perhaps stated the just statement

- Mon Jan 31, 2011 8:58 pm
- Forum: Secondary: Solved
- Topic: Dhaka Secondary 2011/9
- Replies:
**9** - Views:
**4293**

### Re: Dhaka Secondary 2011/9

my approach is this

suppose (x,3)=1 where (a,b) means their gcd

by euler x^2=1 mod 3

and again (2,3)=1

so 2^(2y)=1mod 3

let x=2y

so we need to sort out all those numbers coprime to 3 .but note problem occurs with 6.but in this case PIE (principle of inclusion exclusion) perhaps paves da way

suppose (x,3)=1 where (a,b) means their gcd

by euler x^2=1 mod 3

and again (2,3)=1

so 2^(2y)=1mod 3

let x=2y

so we need to sort out all those numbers coprime to 3 .but note problem occurs with 6.but in this case PIE (principle of inclusion exclusion) perhaps paves da way

- Mon Jan 31, 2011 8:41 pm
- Forum: H. Secondary: Solved
- Topic: Dhaka Higher Secondary 2011/9
- Replies:
**11** - Views:
**6228**

### Re: Dhaka Higher Secondary 2011/9

ups!!

sry.i perhaps misviewed the question.

after all the problem was quite captivating as well as challenging .

thank you..

sry.i perhaps misviewed the question.

after all the problem was quite captivating as well as challenging .

thank you..

- Mon Jan 31, 2011 8:35 pm
- Forum: H. Secondary: Solved
- Topic: Dhaka Higher Secondary 2011/2
- Replies:
**10** - Views:
**4642**

### Re: Dhaka Higher Secondary 2011/2

perhaps you tried its the credit.

- Mon Jan 31, 2011 3:32 pm
- Forum: H. Secondary: Solved
- Topic: Dhaka Higher Secondary 2011/9
- Replies:
**11** - Views:
**6228**

### Re: Dhaka Higher Secondary 2011/9

whats problem with this solution??

1971=S+32+16+2+1

1995=S+64+8+2+1

2011=S+64+16+8+2+1 where S=1024+512+256+128

as f(2^n)=f(2^[n+2])

set f(S)=H(S)

so f(1971)=H(S)+2(f(1)+f(2))

and f(1995)=H(S)+2(f(1)+f(2))

so f(1995)=f(1971)=-1.....

1971=S+32+16+2+1

1995=S+64+8+2+1

2011=S+64+16+8+2+1 where S=1024+512+256+128

as f(2^n)=f(2^[n+2])

set f(S)=H(S)

so f(1971)=H(S)+2(f(1)+f(2))

and f(1995)=H(S)+2(f(1)+f(2))

so f(1995)=f(1971)=-1.....

- Mon Jan 31, 2011 12:59 pm
- Forum: H. Secondary: Solved
- Topic: Dhaka Higher Secondary 2011/2
- Replies:
**10** - Views:
**4642**

### Re: Dhaka Higher Secondary 2011/2

whats problem with 111 and 109.see difference is 02...

- Sun Jan 23, 2011 7:12 am
- Forum: Higher Secondary Level
- Topic: diophantine equation
- Replies:
**4** - Views:
**1483**

### diophantine equation

determine all non negative integers (x,y) for which the relation hold

(xy-7)^2=(x^2+y^2)

indian math olympiad

hint:factorization you all need to hang about

(xy-7)^2=(x^2+y^2)

indian math olympiad

hint:factorization you all need to hang about