# Monthly Archives: January 2015

## Num3ers such xyz + x(10y + z) + (10x + y)z = 100x + 10y + z

for example,   155   Find a 4-digit number abcd such that a*b*c*d   +   abc*d +   ab*cd +     a*bcd   =   abcd     Pipo found : 6550 = 6*5*5*0 + … Continue reading

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## Find sequence of integers: DigitProduct(N)/DigitSum(N)

Consider   3489 the product of its digits divided by the sum of its digits (3*4*8*9)/(3+4+8+9) = 36   is an integer Repeat the process: (3*6)/(3+6) = 2 Stop the process when you end up with a 1-digit … Continue reading

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## Numbers (< 1000) such that concatenation of their prime factors is a prime number

Part 1 Each concatenation is composed by two different digits:

## Numbers such that concatenation of prime factors is a power

For example,

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## Patterns with 1/13

(period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)   (period 6)       … Continue reading

## Patterns with 1/7

(period 6) Move 7 :       (period 6) Move 4 :       (period 6) Move 1 :       (period 6) Move 8 :       (period 6) Move 5 :       (period 6) … Continue reading

## Equations: A^2 + B^3 = C^4 and D^3 + E^4 = F^5

Find positive integer solutions to (1)   (2)     For example,   (1)                                                                                                                                                                                                                                         is a prime number.                                                                                                                                                                                                                                                                                                                      ——————————————   (2)                                … Continue reading

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## Products | (x + r_1)(y + r_2) = xy; (r_1, r_2) are rational

Example #1 :    ,    Integer solutions:    ,    ……………………………………………….. Example #2 :    ,    Integer solutions:    ,    ………………………………………………..   Integer solutions:    ,    ………………………………………………..   there are infinitely many possibilities.   … Continue reading