# Category Archives: Uncategorized

## Num3ers with repeated digits divisible by a prime they contain

Start with a 2-digit prime number 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 and find numbers that repeat the digits of the chosen prime and divisible … Continue reading

## Sum of 7 consecutive num3ers is a cube

Older post:   Sum of 5 consecutive num3ers is a cube   Let   (x-3),   (x-2),   (x-1),   x,   (x+1),   (x+2),   (x+3)   be 7 consecutive integers. (x-3)   +   (x-2)   +   … Continue reading

## Concatenation | Replacing Prime Num3ers by concatenation of their divisors

Let’s consider all the prime numbers under 1000: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, … Continue reading

## Concatenation | N || N = x^2 and N || (N-1) = x(x-2)

Numbers n such that n concatenated with itself is a biperiod square: N || N   =   x^2   is a biperiod square oeis – A102567   1322314049613223140496   =   13223140496||13223140496   =   36363636364^2 2066115702520661157025   … Continue reading

## Integers (base 10) converted as repdigit in 2 or more bases

For example, 40 Base 10   =   1111 base-3   =   55 base-7   =   44 base-9   =   22 base-19   =   11 base-39   Find other examples.   (N.B.   Once you … Continue reading

Posted in Uncategorized | 4 Comments

## Odd Num3ers less than 100 in base 11

Take an odd number less than 100 and multiply it by 95 95 * 1  =  95 95 * 3  =  285 95 * 5  =  475 95 * 7  =  665 95 * 9  =  855 95 * 11 … Continue reading

## Integers k such that the expansion of 2^k in base 10 terminates with k

Answers: k   =   36,   736,   8736,   48736   2^36   =   68719476736 2^736 = 3614737867146518396094859318021923665089733007170019231594754471504248102 8623340798795186188738943961227492678378035156199978199883243404129619879 5326329101623141899709787663433296905279066051548640942013290819886814068736 2^8736 = 6281192747149393281421000011495905489461122106645061916895418841293270371 2485870585757827424131652032338779946099246612887979528006433406017721311 2936407707971981775612404905170935708518283336613810556843698068432864542 8584462664470562248271598748550506873429385403713680400337612267852242584 5032480626410500459893264493975710553283003239346153798655961975661246979 0575660252994558594240532469561907113293466856509564415848643577781574084 9838779743213818321874734876917741646227550108642889721500198654604406782 4401937493861551092584238195040569053592069102288319504974509345580578219 4348092277144758938493901456562667162062087944427963829431124427017852080 9421886017936716240746318649273348751713304893823106655284792227844509373 1488624425751606689018350062245932943947060210484587172816829054402470961 7197864692688253738501804281913746557609127792976366857789907310051633642 3935844280879617503544753551969971237549316599278475426378063257806751983 9697642668146778126401954844669516505470461007917109504864269643049955897 … Continue reading

## LCM, GCD (puzzles)

Definitions: Greatest common divisor http://en.wikipedia.org/wiki/Greatest_common_divisor Least common multiple http://en.wikipedia.org/wiki/Least_common_multiple       (1) Prove that if the LCM of two numbers is equal to the square of their difference,   then their GCD is the product of two consecutive integers. … Continue reading