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There is a number N beginning with 4 such that moving the 4 to the end creates a new number that is a quarter of N. In other words N is of the form 4[…], and N ÷ 4 = […]4. What is the lowest possible value of N?

Start with two digits: N = 4[?]. The only possible [?] is 1 because a quarter of 4 is 1, but 14 is not a quarter of 41, so N has more than two digits. For three digits write N = 41[?]. Since 4 × 1[?]4 = 41[?], the final digit must be 6 because 4 × 4 = 16; however 416 is not 164, so continue.

For four digits let N = 41[?]6. Then 4 × 1[?]64 = 41[?]6, and the penultimate digit must be 5 because 4 × 64 = 256, yet 4156 ≠ 1564. With five digits N = 41[?]56 and 4 × 1[?]564 = 41[?]56; since 4 × 564 = 2256 the antepenultimate digit must be 2, but 41256 ≠ 12564.

Finally, for six digits N = 41[?]256 and 4 × 1[?]2564 = 41[?]256. Because 4 × 2564 = 10256 the [?] must be 0, which gives N = 410256 = 4 × 102564.

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