Question Bank  Diophantine Equations & Auxiliary Lines  Sibanand Pattnaik

Here, I will post some of the most Relevant Questions from Diophantine Equations & Auxiliary Lines. These questions are from amongst the frequently appearing ones in mocks and is a must know type.
Number of Questions  30
Topic  Diophantine Equations & Auxiliary Lines
Answer Key provided  No

Q1) Integers x and y with x > y > 0 satisfy x + y + xy = 80. What is x ?
a) 8
b) 10
c) 15
d) 18
e) 26

Q2) Consider the set of all fractions x/y where x and y are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increase by 1, the value of fraction is increased by 10% ?
a) 0
b) 1
c) 2
d) 3
e) Infinitely many

Q3) A rectangle has area A cm^2 and perimeter P cm, where length and width are positive integers. Which of the following cannot be A + P ?
a) 100
b) 102
c) 104
d) 106
e) 108

Q4) The number of solution pairs in positive integers of the equation 3x + 5y = 501 is
a) 33
b) 34
c) 35
d) 100
e) None of these

Q5) Find the prime number p such that 71p + 1 is a perfect square.

Q6) Find all integer length right triangles with an area equal to its perimeter

Q7) Find all ordered pairs of positive integers (x, y) such that 1/2x + 1/3y = 1/6

Q8) Let n be the number of ways that 10 dollars can be changed into dimes and quarters, with at least one of each coin being used. then n equals:
a) 40
b) 38
c) 21
d) 20
e) 19

Q9) Let n be the number of integer pairs (x, y) which satisfy 5y  3x = 15 and x^2 + y^2 ≤ 25. Then n is
a) 0
b) 1
c) 2
d) 4
e) 6

Q10) The number of positive integers k for which the equation kx  12 = 3k has an integer solution for x is
a) 3
b) 4
c) 5
d) 6
e) 7

Q11) In the xyplane, how many lines whose x intercept is a positive prime number and whose y intercept is a positive integer pass through the point (4, 3)?
a) 0
b) 1
c) 2
d) 3
e) 4

Q12) The number of solutions in positive integers of 2x + 3y = 763 is
a) 255
b) 254
c) 128
d) 127
e) 0

Q13) Penniless Pete's piggy bank has no pennies in it, but it has 100 coints, all nickels, dimes and quarters, whose total value is 8.35 dollars. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
a) 0
b) 13
c) 37
d) 64
e) 83

Q14) For how many ordered pairs of positive integers (x, y) is x + 2y = 100?
a) 33
b) 49
c) 50
d) 99
e) 100

Q15) A line passes through A = (1, 1) and B = (100, 1000). How many other points with integer coordinates are on the line and strictly between A and B?
a) 0
b) 2
c) 3
d) 8
e) 9

Q16) Oscar buys 13 pencils and 3 erasers for 1 dollar. A pencil costs more than an eraser and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser ?
a) 10
b) 12
c) 15
d) 18
e) 20

Q17) The town of Hamlet has 3 people for each horse, 4 sheep for each cow and 3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows and ducks in Hamlet.
a) 41
b) 47
c) 59
d) 61
e) 66

Q18) In how many ways can 1776 identical US flags be partitioned into piles of either three or four so that every flag is in some pile?

Q19) Find the number of ordered triples (a, b, c) where a, b and c are positive integers, a is a factor of b, a is a factor of c, and a + b + c = 100