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In the United Kingdom the currency is made up of pound (\[Sterling]) and \
pence (p). There are eight coins in general circulation:
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It is possible to make \[Sterling]2 in the following way:
1\[Times]\[Sterling]1 + 1\[Times]50p + 2\[Times]20p + 1\[Times]5p + 1\
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How many different ways can \[Sterling]2 be made using any number of coins?\
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We shall say that an n-digit number is pandigital if it makes use of all the \
digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 \
through 5 pandigital.
The product 7254 is unusual, as the identity, 39 \[Times] 186 = 7254, \
containing multiplicand, multiplier, and product is 1 through 9 pandigital.
Find the sum of all products whose multiplicand/multiplier/product identity \
can be written as a 1 through 9 pandigital.
HINT: Some products can be obtained in more than one way so be sure to only \
include it once in your sum.\
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The fraction 49/98 is a curious fraction, as an inexperienced mathematician \
in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which \
is correct, is obtained by cancelling the 9s.
We shall consider fractions like, 30/50 = 3/5, to be trivial examples.
There are exactly four non-trivial examples of this type of fraction, less \
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If the product of these four fractions is given in its lowest common terms, \
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\>", "Text",
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145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.
Find the sum of all numbers which are equal to the sum of the factorial of \
their digits.
Note: As 1! = 1 and 2! = 2 are not sums they are not included.\
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The number, 197, is called a circular prime because all rotations of the \
digits: 197, 971, and 719, are themselves prime.
There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, \
73, 79, and 97.
How many circular primes are there below one million?\
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"The decimal number, 585 = ",
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" (binary), is palindromic in both bases.\nFind the sum of all numbers, less \
than one million, which are palindromic in base 10 and base 2.\n(Please note \
that the palindromic number, in either base, may not include leading zeros.)"
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The number 3797 has an interesting property. Being prime itself, it is \
possible to continuously remove digits from left to right, and remain prime \
at each stage: 3797, 797, 97, and 7. Similarly we can work from right to \
left: 3797, 379, 37, and 3.
Find the sum of the only eleven primes that are both truncatable from left to \
right and right to left.
NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.\
\>", "Text",
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Cell["\[FilledSquare]38. Pandigital multiples", "Subsubsection",
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Take the number 192 and multiply it by each of 1, 2, and 3:
192 \[Times] 1 = 192
192 \[Times] 2 = 384
192 \[Times] 3 = 576
By concatenating each product we get the 1 to 9 pandigital, 192384576. We \
will call 192384576 the concatenated product of 192 and (1,2,3)
The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, \
and 5, giving the pandigital, 918273645, which is the concatenated product of \
9 and (1,2,3,4,5).
What is the largest 1 to 9 pandigital 9-digit number that can be formed as \
the concatenated product of an integer with (1,2, ... , n) where n > 1?\
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If p is the perimeter of a right angle triangle with integral length sides, \
{a,b,c}, there are exactly three solutions for p = 120.
{20,48,52}, {24,45,51}, {30,40,50}
For which value of p \[LessEqual] 1000, is the number of solutions maximized?\
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