<p>Find the smallest natural number \(n\) whose last digit is 6, and if the last digit is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number.</p>
Step-by-Step Solution
Key Concept: Set up an algebraic equation by expressing the original number as 10a + 6 (where a represents all digits except the last), then express the rearranged number as 6×10^k + a. The condition that the rearranged number equals 4 times the original gives us a solvable equation.
<p><strong>Step 1:</strong> Let the original number be represented as <strong>a6</strong> (where a is a string of digits), which equals 10a + 6 in value.</p><p><strong>Step 2:</strong> When the last digit 6 is moved to the front, we get <strong>6a</strong>. If a has k digits, this number equals 6×10^k + a.</p><p><strong>Step 3:</strong> According to the problem: 6×10^k + a = 4(10a + 6)</p><p><strong>Step 4:</strong> Simplifying: 6×10^k + a = 40a + 24</p><p>6×10^k = 39a + 24</p><p>6(10^k - 4) = 39a</p><p>a = 6(10^k - 4)/39 = 2(10^k - 4)/13</p><p><strong>Step 5:</strong> For a to be a natural number, (10^k - 4) must be divisible by 13.</p><p><strong>Step 6:</strong> Testing values of k:</p><p>• k = 1: 10 - 4 = 6 (not divisible by 13)</p><p>• k = 2: 100 - 4 = 96 (not divisible by 13)</p><p>• k = 3: 1000 - 4 = 996 = 13×76 + 8 (not divisible)</p><p>• k = 4: 10000 - 4 = 9996 = 13×769 + 7 (not divisible)</p><p>• k = 5: 100000 - 4 = 99996 = 13×7692 ✓</p><p><strong>Step 7:</strong> When k = 5: a = 2(99996)/13 = 2(7692) = 15384</p><p><strong>Step 8:</strong> The original number is 153846.</p><p><strong>Verification:</strong> Moving 6 to front gives 615384 = 4×153846 ✓</p><p><strong>∴ Answer: 153846</strong></p>
Correct Answer: 153846