10,000,000--10,000,999 (separate page)

10,001,628

[Math.]  A triangular number (sum of all integers from 1 to 4472):

10,001,628 = 1+2+…+4471+4472.

 

10,004,569

[Math.]  The smallest 8-digit square number = 3163².

 

10,007,820

[Math.]  Two consecutive sums of consecutive numbers: (smallest 8-digit)

46,225+…+46,440 = 46,441+…+46,655 = 10,007,820.

 

10,025,290

[Math.]  A pair of amicable numbers (9660950, 10025290), one of 8 pairs involving a 7-digit and an 8-digit number.

 

10,036,224

[Math.]  3168² = 10,036,224 and, reversely, 42,263,001 = 6501².

 

10,043,690

[Math.]  A pair of amicable numbers (8826070, 10043690), one of 8 pairs involving a 7-digit and an 8-digit number.

 

10,075,156

[Math.]  A pyramidal number (sum of all squares of integers from 1 to 311):

10,075,156 = 1²+2²+…+310²+311².

 

10,077,696

[Math.]  The smallest 8-digit cube number, 10,077,696  = 216³.

 

10,200,960

[Math.]  The order of the 6^th sporadic group: Mathieu group M₂₃.

 

10,234,567

[Math.]  The smallest number of 8 different digits.

 

10,234,589

[Math.]  The smallest prime number of 8 different digits.

 

10,273,670 & 10,254,970

[Math.]  A pair of amicable numbers (10254970, 10273670), the smallest 8-digit pairs.

 

10,481,761

[Math.] 

555533333 is divisible by 53: 555533333 = 53×10481761.

44333 is divisible by 43: 44333 = 43×1031.

22333 is divisible by 23: 22333 = 23×971.

 

10,497,600

[Math.]  Sum of the first 80 cube numbers:

1³+2³+…+79³+80³ = 10,497,600 = (1+2+…+79+80)² = 3240².

 

10,510,564

[Math.]  3242² = 10,510,564.

 

10,533,296

[Math.]  A pair of amicable numbers (10533296, 10949704).

 

10,572,550

[Math.]  A pair of amicable numbers (10572550, 10854650).

 

10,596,368

[Math.]  A pair of amicable numbers (10596368, 11199112).

 

10,600,000s--10,900,000s (separate page)

 

xy,abc,def

[Math.] For any xy = 10 to 99, there are always six 6-digit numbers abc,edf such that the sum (xy+abc,def) is equal to one of 6 digit-rotations of 142,857 and the number xy,abc,def is a multiple of 142,857.

 

For any xy = 10 to 99, there are always twelve 6-digit numbers abc,edf such that the sum (xy+abc,def) is equal to one of 12 digit-rotations of 076,923 or 153,846 and the number xy,abc,def is a multiple of 76,923.