Question

Let m and n be positive integers satisfyingmri2 + 876 = 4mn + 217n.Find the sum of all possible values of m. correct answer needed​

Answer 1

SOLUTION

GIVEN

Let m and n be positive integers satisfying mn² + 876 = 4mn + 217n

TO DETERMINE

The sum of all possible values of m

EVALUATION

Here it is given that

$\displaystyle\sf{m {n}^{2} + 876 = 4mn + 217n}$

$\displaystyle\sf{ \implies \: m {n}^{2} - 217n = 4mn - 876}$

$\displaystyle\sf{ \implies \: n(mn- 217) = 4mn - 876}$

$\displaystyle\sf{ \implies \: n = \: \frac{(4mn - 876)}{(mn- 217)} }$

$\displaystyle\sf{ \implies \: n = \: \frac{4(mn - 217) - 8}{(mn- 217)} }$

$\displaystyle\sf{ \implies \: n = \: 4 - \frac{ 8}{(mn- 217)} } \: \: - - - (1)$

Since n is an integer

So mn - 217 must divide 8 completely

Thus we get

$\displaystyle\sf{ \implies(mn- 217) = \pm \:1 \:, \: \pm \:2 \:, \: \pm \:4 \:, \: \pm \:8}$

$\displaystyle\sf{ \implies \: mn = 218,216,219,215,221,213,225,209}$

From Equation 1 we get

$\displaystyle\sf{ \implies \: n = - 4,12,0,8,2,6,3,5}$

Therefore there are only two pair of values of ( mn , n ) precisely ( 216 , 12 ) , ( 225 , 3 )

Consequently the values of m are 18 and 75 respectively

Thus sum of all possible values of m

= 18 + 75

= 93

FINAL ANSWER

The sum of all possible values of m = 93

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Use Euclid's algorithm to find the HCF of

(1) 900 and 270

(ii) 196 and 38220

Use division

https://brainly.in/question/32525059

2. Find the HCF of 315 and 600 by using Euclid's division algorithm.

https://brainly.in/question/38101510

Answer 2

Answer:

$\underline{\purple{\ddot{\Maths dude}}}$

◇◇Given:-

• Let m and n be the positive integers satisfying mn^2+876=4mn+217n

◇◇To prove :-

• Find the sum of all possible values of m .

◇◇Explanation :-

• In the question given that,

1. mn^2+876=4mn+217n

• mn^2-217n=4mn-876

• n (mn-217)=4mn-876

• n= (4mn-876)/(mn -217)

• n= (4mn-217)-8/(mn-217)

• n=4 - 8 /(mn-217) (i)

Here n is an integer ,

• Here mn-217mist divide by 8 perfectly and completely

Therefore,

we get,

(mn-217)= +/- 1, +/-2, +/- 4, +/-8.

• mn = 218,216,219,215,221,213,225,209.

From the equation we know,

• n=-4,12,0,8,2,6,3,5.

◇◇Hence , here there are only two pairs of integers they are ( 216,12),(225,3)

• Here the values m are 18 and 75 respectively ,

◇◇According to question,

• we get,
• Sum of all possible of m
• =18+74
• =93.

◇◇Hope it helps u mate .

♧♧Thank you.