Question

The number of positive integral solutions of 2x + 3y = 763 is

(A) 125 (B) 126

(C) 127 (D) 12

Answer 1

Concept: In general, the word "integral value" refers to the result of integrating or summing the terms of a function that has an infinite number of terms.

Given: 2x + 3y = 763

To find: The number of positive integral solutions

Solution: 2x + 3y = 763

y = (763 - 2x)/3

Since there are infinitely many integral values of "x" for which "y" is an integer, it has infinitely many integral solutions.

Now, (763-2x) must be a multiple of 3 in order for "y" to be an integer.

It occurs when

763 - 2x = 759, 753, 747, 741, 735, 729.....15, 9, 3 and

x  = 2, 5, 8, 11, 14, 17, ........., 371, 374, 377, 380, according to a small observation.

We can calculate y for any value of "x".

Since the values of "x" make up an A.P., therefore, by using the formula

$a_{n} = a + (n - 1) d$  ;

where $a_{n}$ is the nth term of an AP,

            a is the first term,

            n is the total number of terms in an AP

            d is the common difference

380 = 2 + (n-1) * 3

380 = 2 + 3n - 3

380 = 3n -1

380 + 1 = 3n

381 = 3n

n=127

As a result, the equation has 127 solutions.

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Answer 2

Concept: In general, the word "integral value" refers to the result of integrating or summing the terms of a function that has an infinite number of terms.

Given: 2x + 3y = 763

To find: The number of positive integral solutions

Solution: 2x + 3y = 763

y = (763 - 2x)/3

Since there are infinitely many integral values of "x" for which "y" is an integer, it has infinitely many integral solutions.

Now, (763-2x) must be a multiple of 3 in order for "y" to be an integer.

It occurs when

763 - 2x = 759, 753, 747, 741, 735, 729.....15, 9, 3 and

x  = 2, 5, 8, 11, 14, 17, ........., 371, 374, 377, 380, according to a small observation.

We can calculate y for any value of "x".

Since the values of "x" make up an A.P., therefore, by using the formula

 ;

where  is the nth term of an AP,

           a is the first term,

           n is the total number of terms in an AP

           d is the common difference

380 = 2 + (n-1) * 3

380 = 2 + 3n - 3

380 = 3n -1

380 + 1 = 3n

381 = 3n

n=127

As a result, the equation has 127 solutions.