math grad - Interest: Number theory

Using 9,9,9,9 to make 1-100

Using   9, 9, 9, 9   and the basic operations make the numbers   1   to   100. You may also use decimal points.   Here are the numbers   1   to   20:   1   … Continue reading

Triangle (5, 7, 8)

A triangle with sides of length   5,   7,   and   8 Perimeter   =   20 Area   =   10 √3   ~   17.3205 one of its angles measures 60 degrees: Applying the Law … Continue reading

Primes p; n(p + n) is a square

Consider the first few prime numbers:           2,   3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37 And   f(p) = n(p + n),     where  p  is an odd prime and … Continue reading

Using 1,2,3,4,5 only once to make 1 – 100

Game: Using  1, 2, 3, 4, 5  only once and the basic operations (+, -, * , / , !) to make all the numbers from  1  to  100   0 = 5 + 3 – (4*2/1)       … Continue reading

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

Concatenation | (x – y + z)^3 = x || y || z

Older post:   Concatenation: x||y||z = x^3 + y^3 + z^3   Find integers   x,   y   and   z   such that                                                  (x – y + z)^3   =   x || y || z                                                  … Continue reading

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Using Primes 2,3,5,7 only once to make 1-100

Game: You can use the primes   2, 3, 5, 7   and the basic operations   +, -, *, /   and   !   (factorial) only once to make numbers from   1   to   100.   … Continue reading

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Non-palindrome multiplied by its reverse is a square

For example, 288   is a non-palindrome and a non-square number. Its reverse is:     Rev (288)   =   882 and, 288   *   882   =   254016   =   504^2   is a perfect … Continue reading

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DigitProduct and Sum of prime factors of numbers

(1) Integers such that the sum of prime factors equal the product of the digits of those integers:   The prime factors of   18   are:                           … Continue reading