12345679 times 9, and its friends

Question

Take the number 12,345,679, which is the digits 1 to 9 in order with the 8 left out. What is 12,345,679 × 9? What are 12,345,679 × 18, × 27 and × 45? Explain why the pattern happens, and say where the missing 8 went.

Accepted Answer

12,345,679 × 9 = 111,111,111. Multiplying by 18, 27 and 45 gives 222,222,222, 333,333,333 and 555,555,555. Why. 111,111,111 = 9 × 12,345,679, which you can verify by long division: 9 goes into 11 once with 2 left, into 21 twice with 3 left, into 31 three times with 4 left, and so on; the quotient's digits count up 1, 2, 3, 4, 5, 6, 7, and then 9 into 71 goes 7 times with 8 left, and 9 into 81 goes 9 times exactly, which is why the 8 is skipped and the quotient ends 7, 9. So 12,345,679 is exactly one ninth of the nine-digit repunit. The multiples. 12,345,679 × 9k = k × 111,111,111 = the digit …

12345679 times 9, and its friends

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