Properties of numbers – sevens

This chapter will undoubtedly be the most interesting, most thrilling for these, who fell under the spell of the number. And the charm is peculiar indeed, but having some resemblance to the charm of music, colors, dance, to the spell of a living word, to the spell of poetry at all. And like almost everyone in his youth, he was to a greater or lesser extent a poet, so almost everyone has succumbed and succumbed to the charm of the number.

Of course, this chapter does not cover all the rich material in this area of ​​issues. But this kind of "continuation" will be found in the second volume, but with this difference, that there will be no need to return to the interesting properties of many of the numbers quoted here. However, there will be new ones, different, yet full of charm …

A strange seven

When arithmetic progress, whose first word and difference is a number 15 873, we will multiply by 7, we get very strange products. Numbers 15 873, 31 746, 47 619, 63 492, 79 365, 95 238, . . ., 142 857, multiplied by 7, will always give a composite number z 6 times repeated the same digit:

15873 • 7 = 111111
31746 • 7 = 222222
……………….
79365 • 7 = 555555
……………….
142857 • 7 = 999999

This interesting combination of numbers can be easily explained, when we notice, that, for example

79365 • 7 = (5-15873) • 7 = 5 • (15873 • 7) = 5 • 111111

This unusual phenomenon is more difficult to explain, that if between two digits of the second power of a number 7, that is, in the middle of the number 49 we're going to insert a number 48, are the numbers formed this way, namely:

they will always be full squares:

49 = 7²
4489 = 67²
444889 = 667²
44448889 = 6667²

But even more interesting "wonders” can be obtained from a combination of numbers 7 with numbers 11 i 13 or - if you prefer - with a number 143 equal 11 • 13.

Well, if we multiply the number 143 by any of the 999 prime in natural order of multiples of a number 7, then in the product we always get a number composed of two identical numbers, e.g:

28 • 143 = 4004
315 • 143 = 45045
2464 • 143 = 352352
3591 • 143 = 513513
5495 • 143 = 785785
6993 • 143 = 999999

And it should be noted, that the number repeating in the product is always equal to the number of sevens in the multiplier. In fact:

28:7 = 4
315 :7 =45
2464 : 7 = 352
and so on.

This seemingly strange phenomenon is explained very simply. Just say, that 7 • 143 = 1001. Because
2464 • 143 = (352 • 7) • 143 = 352 • (7 • 143) = 352 • 1001 =
= 352 • 1000 + 352 = 352352

Similar results are obtained by multiplying 77 by 999 first multiples of a number 13 or also by multiplying 91 by 999 first multiples of a number 11.