Number 36 in Math

A page of the Numeropedia - the Special Encyclopedia of Numbers

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A triangular number (sum of all integers from 1 to 8): 36 = 1+2+3+4+5+6+7+8. The pair 15 and 21 is the smallest pair of triangular numbers whose sum (36) and difference (6) are also triangular.

The 2nd square number (62) that is also a triangular number. The nth root number is 6 times the (n–1)th root minus the (n–2)th root: 1, 62, 352, 2042, 11892

362 = (1+29+6)2 = 1,296.

364 = (1+6+7+9+6+1+6)4 = 1,679,616.

365 = (6+0+4+6+6+1+7+6)5 = 60,466,176.

Three square numbers using the same digits: 362 = 1,296, 542 = 2,916 and 962 = 9216.

A base system of multigrade equalities:

1+8+10+17 = 36 = 2+5+13+16

12+82+102+172 = 454 = 22+52+132+162

13+83+103+173 = 6,426 = 23+53+133+163. (

Multigrade equalities still hold by adding any integer n to every base term in both sides.  Hence, the base equality always has number 1 appearing in one side.

Sum of the first 3 cube numbers: 13+23+33 = 36 = (1+2+3)2.

The isosceles triangle with 3 angles (36o, 72o, 72o) is called a golden triangle, since its sides possess the golden ratio ½(1+51/2).

62 = 36 and if all digits are decremented (except the exponent) by 1: 52 = 25.

Repeatedly inserting numbers 36 in the middle of the prime number 11 yields other prime numbers: 1361, 136,361, 13,636,361, 1,363,636,361 and 136,363,636,361.

12×21 = 252 = 21×12

12×42 = 504 = 21×24          24×21 =    504 = 42×12

12×63 = 756 = 21×36          24×63 = 1,512 = 42×36

12×84 = 1,008 = 21×48       24×42 = 1,008 = 42×24

36×84 = 3,024 = 63×48       24×84 = 2,016 = 42×48.

 33×34 = 1122 33×35 = 1155 33×36 = 1188 33×37 = 1221 33×47 = 1551 33×57 = 1881

 333×334 = 111,222 333×335 = 111,555 333×336 = 111,888 333×337 = 112,221 333×347 = 115,551 333×357 = 118,881

… (generalizable pattern) …

987,654,321×09 = 08,888,888,889

987,654,321×18 = 17,777,777,778

987,654,321×27 = 26,666,666,667

987,654,321×36 = 35,555,555,556

987,654,321×45 = 44,444,444,445

987,654,321×54 = 53,333,333,334

987,654,321×63 = 62,222,222,223

987,654,321×72 = 71,111,111,112

987,654,321×81 = 80,000,000,001.

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References [26]