BMAD 65 Posted April 18, 2013 Report Share Posted April 18, 2013 Find the smallest positive integer such that if the ones digit is moved (from the right) all the way to the left, the resulting number is exactly 50% more than the original number. Quote Link to post Share on other sites

0 Solution kevink2 0 Posted April 18, 2013 Solution Report Share Posted April 18, 2013 is 285714 Quote Link to post Share on other sites

0 TimeSpaceLightForce 12 Posted April 18, 2013 Report Share Posted April 18, 2013 6digits? 5 7 1 4 2 8 -140000 -14000 -1400 -140 -14 99998.5 -700000 -98000 -1400 -560 -28 799988 = 0 Quote Link to post Share on other sites

0 kalaivanan 0 Posted April 18, 2013 Report Share Posted April 18, 2013 The smalles positive integer is 2143 its 50% is 1071.5 result number is 3214.5 Quote Link to post Share on other sites

0 vinay.singh84 0 Posted April 18, 2013 Report Share Posted April 18, 2013 Let's the value be m and its shifted version n. 2/3*n = m m and n are both integers, so n-m=1/3*n is also an integer. Therefore 3 | n . As their digits have the same sum, 3 | m. Therefore 3 | n-m --> 3 | 1/3*n --> 9 | n. As their digits have the same sum, 9 | m. Following similar logic from above, above 9 | n-m --> 9 | 1/3*n --> 27 | n. So you need only account for 1/27 of the integers for a brute strength solution (which speeds it up significantly) I got 285714 and 571428. Quote Link to post Share on other sites

0 BMAD 65 Posted April 18, 2013 Author Report Share Posted April 18, 2013 Let's the value be m and its shifted version n. 2/3*n = m m and n are both integers, so n-m=1/3*n is also an integer. Therefore 3 | n . As their digits have the same sum, 3 | m. Therefore 3 | n-m --> 3 | 1/3*n --> 9 | n. As their digits have the same sum, 9 | m. Following similar logic from above, above 9 | n-m --> 9 | 1/3*n --> 27 | n. So you need only account for 1/27 of the integers for a brute strength solution (which speeds it up significantly) I got 285714 and 571428. brilliant observation Quote Link to post Share on other sites

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