Question 860872
Let the integers be {{{ a }}} and {{{ b }}}
Assume that {{{ a }}} is the larger
(1) {{{ a - b = 8 }}}
(2) {{{ 1/b - 1/a = 1/6 }}}
Note that 1 / ( small number) is greater than 1 / ( big number )
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(2) {{{ 1/b - 1/a = 1/6 }}}
Multiply both sides by {{{ 6*a*b }}}
(2) {{{ 6a - 6b = a*b }}}
(2) {{{ 6a - a*b = 6b }}}
(2) {{{ a*( 6 - b ) = 6b }}}
and
(1) {{{ a = 8 + b }}}
By substitution:
(2) {{{ ( 8 + b )*( 6 - b ) = 6b }}}
(2) {{{ 48 + 6b - 8b - b^2 = 6b }}}
(2) {{{ -b^2 - 2b - 48 = 6b }}}
(2) {{{ b^2 + 8b - 48 = 0 }}}
(2) {{{ ( b + 12 )*( b - 4 ) = 0 }}}
{{{ b = 4 }}} ( choose the positive root )
and
(1) {{{ a = 8 + b }}}
(1) {{{ a = 8 + 4 }}}
(1) {{{ a = 12 }}}
The integers are 4 and 12
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check:
(2) {{{ 1/b - 1/a = 1/6 }}}
(2) {{{ 1/4 - 1/12 = 1/6 }}}
(2) {{{ 3/12 - 1/12 = 2/12 }}}
(2) {{{ 2/12 = 2/12 }}}
OK