Reciprocal of 61
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Theorem
The decimal expansion of the reciprocal of $61$ is as follows:
- $\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$
This sequence is A021065 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.01639344262295081967213114754098360655737704918032786885245901...
--------------------------------------------------------------------
61)1.00000000000000000000000000000000000000000000000000000000000000000
61 122 61 61 488 183 183 305
-- --- --- -- --- --- --- ---
390 380 590 90 220 470 170 150
366 366 549 61 183 427 122 122
--- --- --- -- --- --- --- ---
240 140 410 290 370 430 480 280
183 122 366 244 366 427 427 244
--- --- --- --- --- --- --- ---
570 180 440 460 400 300 530 360
549 122 427 427 366 244 488 305
--- --- --- --- --- --- --- ---
210 580 130 330 340 560 420 550
183 549 122 305 305 549 366 549
--- --- --- --- --- --- --- ---
270 310 80 250 350 110 540 100
244 305 61 244 305 61 488 61
--- --- -- --- --- --- --- ---
260 500 190 600 450 490 520 ...
244 488 183 549 427 488 488
-- --- --- --- --- --- ---
160 120 70 510 230 200 320
122 61 61 488 183 183 305
--- --- -- --- --- --- ---
$\blacksquare$