This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1849 Excerpt: ...of it j multiply it then by such a number as will make the coeff. of y div. by the den" with rem' 1, i. e. in this case, mult, it by 2; 2-4y 2-y.. A 2-v.... then--. or---y is int., .'.--5.. is int. = t suppose; 00 o hence 2-y = 3, or y = 2-3, and a: = i(73-5y) = 21 + 5t. Thus, if we take t = 0, then x = 21, y-2; if t=l ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1849 Excerpt: ...of it j multiply it then by such a number as will make the coeff. of y div. by the den" with rem' 1, i. e. in this case, mult, it by 2; 2-4y 2-y.. A 2-v.... then--. or---y is int., .'.--5.. is int. = t suppose; 00 o hence 2-y = 3, or y = 2-3, and a: = i(73-5y) = 21 + 5t. Thus, if we take t = 0, then x = 21, y-2; if t=l, x = 26, y =-l; if t=-l, x = 16, y = 5; &c. If we require only positive integral values of x and y, then we cannot take t positively, nor therefore 0, or negatively nor therefore 4; hence the values for t range from-4 to 0 inclusively, and thus there will be only 5 positive integral solutions. N.B. It may be shewn that it is always possible to find such_ a number for multiplier as we have employed above, which shall be less than the denominator: and this is the reason why we divide by the least of the two coefficients, in order to have the multiplier as low as possible. But when the denominators are both large, a little ingenuity will save the trouble of searching for such a number, by some such reasoning as that in the next Ex., it being noticed, that the point to be aimed at is, to get the coefficient of y (or of x, as the case may be) in the numerator to be unity. Ex. 2. Solve in positive integers 39x-56y = 11. ., 17y+ 11..-, 34y + 22 Here x-y- y = i...----is int., and...----, 34y + 22 5--22, 40y-176, y-20 let = t;.: y = 39t + 20, and x = s (11 + 56y) = 56 + 29, If we take t-0, then x = 29, y-20, which are the least positive integral values they admit of: but the number of such values is here unlimited, since we may take any positive value for t. 6. In how many ways could 12 guineas be made up of half. guineas and half.crowns? In how many ways, of guineas and crowns? 7. How many fractions are ther...
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