If (a² + 2ab -3b²) =61, where 'a' and 'b' are natural numbers, then what are the values of 'a' and 'b'?

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[math]\\[/math] [math]a^2 + 2ab - 3b^2 = 61 [/math] [math]\implies (a …

Mathematics for School & Entrance Level • 1.5K followers This space shows mathematics, suitable for school & Engineering Entrance Exam.   If (a² + 2ab -3b²) =61, where 'a' and 'b' are natural numbers, then what are the values of 'a' and 'b'? Ragu Rajagopalan, Passionate Maths solver ;Reviving knowledge after 3 decades Answered March 13   [math]\\[/math] [math]a^2 + 2ab - 3b^2 = 61 [/math] [math]\implies (a + 3b) \cdot (a - b) = 61 \tag {E01} \\[/math] [math]\text{a,b are natural numbers, RHS is prime and RHS > 0: } [/math] [math]\qquad \implies a > b; \quad a \neq b ; \quad a,b > 0 \\[/math] [math]\text{Considering the factors of 61 : } [/math] [math]\text{Case - 1 : When } \: \begin{cases} (a+3b) = 61 \\ (a-b) = 1 \end{cases} \: : [/math] [math]\qquad \boxed{\boxed{\: a = 16; \quad b = 15 \:}} \\[/math] [math]\text{Case - 2 : When } \: \begin{cases} (a+3b) = 1 \\ (a-b) = 61 \end{cases} \: : [/math] [math]\qquad b = -15 \: \ngtr 0 \: \times \\[/math] Read more »     If (a² + 2ab -3b²) =61, where 'a' and 'b' are natural numbers, then what are the values of 'a' and 'b'? Doug Dillon, Ph.D. Mathematics Answered March 17   As a quadratic in a, [math]a^2+(2b)a-3b^2–61=0[/math] whose discriminant must be a square and so [math]16b^2+244[/math] is a square .In other words, b=15. and then [math]a^2+30a-736=0[/math] so that [math]a=16.[/math] Read more »     If (a² + 2ab -3b²) =61, where 'a' and 'b' are natural numbers, then what are the values of 'a' and 'b'? Priyanshu Singh Rawat Answered March 13   [math]a^2+2ab-3b^2=61[/math] [math]a^2-b^2+2ab-2b^2=61[/math] [math](a-b)(a+b+2b)=61[/math] [math](a-b)(a+3b)=1×61 [/math] Obviously for natural numbers [math]a[/math] and [math]b[/math], [math]a-b<a+3b[/math]. [math]a-b=1[/math] [math]a+3b=61[/math] By solving pair linear equations. [math]\boxed{a=16, b=15}[/math] Read more »   Read more in Mathematics for School & Entrance Level

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