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- #1

#### Albert

##### Well-known member

- Jan 25, 2013

- 1,225

$a,b\in N$

given:

$(3a+b)^2+6a -2b =1544$

find:

$a+b=?$

given:

$(3a+b)^2+6a -2b =1544$

find:

$a+b=?$

- Thread starter Albert
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- Thread starter
- #1

- Jan 25, 2013

- 1,225

$a,b\in N$

given:

$(3a+b)^2+6a -2b =1544$

find:

$a+b=?$

given:

$(3a+b)^2+6a -2b =1544$

find:

$a+b=?$

- Mar 31, 2013

- 1,358

$a,b\in N$

given:

$(3a+b)^2+6a -2b =1544$

find:

$a+b=?$

add 4b+1 to both sides to get

$(3a+b)^2 + 2(3a+b) + 1 = 1545 + 4b$

or $(3a + b + 1)^2 = 1545 + 4b$

as 1545 mod 4 = 1 solution may exist

so we need to take odd squares above 1545

$(3a + b + 1) = 41 => 1545 + 4b = 1681 => b= 34$

this gives a = 2 or a + b = 36

$(3a + b + 1) = 43 => 1545 + 4b = 1849 => b= 76$ too large

so a+b = 36 is the only solution

- Thread starter
- #3

- Jan 25, 2013

- 1,225

nice solution

add 4b+1 to both sides to get

$(3a+b)^2 + 2(3a+b) + 1 = 1545 + 4b$

or $(3a + b + 1)^2 = 1545 + 4b$

as 1545 mod 4 = 1 solution may exist

so we need to take odd squares above 1545

$(3a + b + 1) = 41 => 1545 + 4b = 1681 => b= 34$

this gives a = 2 or a + b = 36

$(3a + b + 1) = 43 => 1545 + 4b = 1849 => b= 76$ too large

so a+b = 36 is the only solution