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Lecture 3

School

University of WaterlooDepartment

MathematicsCourse Code

MATH135Professor

Roxane Itier ItierLecture

3This

**preview**shows half of the first page. to view the full**1 pages of the document.**MATH 135, Winter 2015

Exercises: Linear Diophantine Equations

1. Decide whether the following linear Diophantine equations admit solutions (without solving

them).

(a) 4x7y 100

(b) 42x56y21

(c) 6x22y18

2. Find a general solution to the following linear Diophantine equations.

(a) 6x4y60

(b) 27x78y12

(c) 81x24y6

3. For each equation in the previous question, ﬁnd all positive solutions, if any exist.

4. When Mr. Smith cashed a cheque for xdollars and ycents, he received instead ydollars and

xcents, and found that he had two cents more than twice the proper amount. For how much

was the cheque written?

5. (Hundred Fowls Problem) Roosters costs 5 qian (ancient Chinese coins) each, hens 3 qian

each and three chickens cost 1 qian. If 100 fowls are bought for 100 qian, how many roosters,

hens and chickens are there? Find all positive solutions.

1

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