Monthly Archives: September 2012

Consecutive numbers each divisible by a cube

(1)   Two consecutive numbers each divisible by a cube. (2)   Three consecutive numbers each divisible by a cube. (3)   the smallest set of numbers of  4, 5, 6, 7   consecutive numbers each divisible by a cube. (1) 80   … Continue reading

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Curious properties of 11, 111, 1111, …

Older post:   Series of Num3ers 1111…     11   is a prime number. 111 = 3 * 37 1111 = 11 * 101 11111 = 41 * 271 ….. 1111111111111111111   (19 digits)   is a prime number. … Continue reading

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Squares with the same digits

A study of square numbers up to 10^5 has revealed that …     … there is only one set of six squares that have the same digits:           103^2 = 10609     130^2 = 16900     140^2 = 19600 … Continue reading

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Integer with 12 and 16 representations as a sum of 2 squares

160225   has 12 representations as a sum of 2 squares: It is also the smallest number that can be written as sum of two perfect squares in 12 ways. 160225 = =   15^2   +   400^2 = … Continue reading

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Integer with 2 representations as a sum of 2 positive 4-th powers

That is to say,   N   =   a^4   +   b^4   =   c^4   +   d^4 where N, a, b, c, and d are all different, non-zero, positive integers   This is a problem … Continue reading

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Letter sums Part 1

Letters represent numbers 0 – 9. Add horizontally and add vertically             HCA   +   ZAA   =   BKZ         CCL   +   CAK   =   HRC         ___________________         … Continue reading

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Squares where all digits are incremented by 1

    Adding 1 to each of their digits means   thus obtaining the repunits   1,   11,   1111,   11111,   1111111,   …                    

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3 Amazing 12-digit numbers

100307124369,   111824028801,   433800063225 They are all square numbers: 100307124369   =   316713^2 111824028801    =   334401^2 433800063225   =   658635^2 The product is: 100307124369 * 111824028801 * 433800063225 = 69755469001138755^2 Sum of the digits: 1+0+0+3+0+7+1+2+4+3+6+9 … Continue reading

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Sequential digits in Sums

Digits are in sequence (not necessarily in order) in the following sums: For example: 1   +   2   =   3     (1, 2, 3) 4   +   5   =   3^2     (2, 3, 4, … Continue reading

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Sums using digits 1 to 9 only once

All digits 1-9 are used once in each sum:     173   +   286   =   459 173   +   295   =   468 127   +   359   =   486 127   … Continue reading

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