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This essential trains for: SAT-I, GMAT, AMC-8, Math Kangaroo 7-8, Math Kangaroo 9-10.
Diophantine equations are equations that involve only integer numbers.
Since the numbers must be integer, Diophantine equations may have more than one unknown each. The constraint that the numbers must be integer may provide additional clues needed to find a solution.
Few Diophantine equations are solvable. Most of them are not. Unlike, for instance, linear equations, they are not guaranteed to have solutions.
Diophantine systems of equations may be written. They have the same properties as the equations.
How do we recognize Diophantine equations and systems?
If the problem clearly states that the quantities involved are integers, or digits, then the equation is Diophantine.
Example: What is the sum of the smallest positive integers x and y that satisfy:
Obviously, since the prime factorization of a positive integer is unique, and since 2 and 5 are prime, we must have identical prime factorizations on both sides of the equation. Therefore:
and the answer is 25+8=33.
These are the smallest possible numbers that satisfy the equation. Of course, there is an infinity of larger numbers that satisfy it, provided we add the same extra factors to both x and y, as in:
Another way of stating the problem is:
What is the sum of x and y, coprime (mutually prime), that satisfy:
If we say that x and y are coprime, then they are implicitly integer and have no common factors
Diophantine equations may otherwise be concealed into word problems that involve quantities that are inherently positive integers: children, stars, atoms, octopi, etc.
In McGrew's Zoo, a fluffinator costs 1 Ylanc a day for his maintenance while a parascream costs 5 Ylancs a day. If Johnny McBonnie donates 156 Ylancs, for how many fluffinators and how many parascreams would this amount support for some days? (characters from Dr. Seuss' "If I Ran the Zoo")
Solution Denote with F the number of fluffinators and with B the number of parascreams. Each day, these creatures' maintenance cost is:
Over N days, the amount needed is:
Since the F, B and N can only be positive integers, let us factor the right hand side:
and use the fundamental theorem of arithmetic: the prime factorization of a number is unique, therefore the left hand side must have the same factorization as the right hand side.
Follows case-work. Case 1:
In this case, the amount can support 4 fluffinators and 1 parascream for 17 days.
In this case, for 3 days, 9 possible combinations of fluffinators and parascreams can be found:
For 9 days, the possible combinations of creatures are:
Note that, for this problem, there is a finite number of solutions.
Such problems may ask how many possible solutions there are or may incorporate additional information that allows to reduce the number of solutions.
Diophantine equations and word problems that yield Diophantine equations and systems of equations are based on an incredibly varied pool of mathematical facts, challenging the creativity and depth of knowledge of each student.
Often, Diophantine equations require using: parity analysis, modular arithmetic, Euclid's algorithm and other number theoretical tools. The range of difficulty of such problems starts at SAT I level and goes all the way to the IMO - therefore, expect them to be one of the workhorses of the examiner!