Question

**Integers and Diophantine Equations Homework Help**

1) Can 120 be expressed as the sum of two positive integers, one of which is divisible by 11 and the other by 17?

2) The nickel slot of a pay phone will not accept coins. Can a call costing 95 cents be paid for exactly using only dimes and quarters? If so, in how many ways can it be done?

Answer 1

1)we cannot express $120$ as the sum of two positive integers, one of which is divisible by $11$ and the other by $17.$

2)There are two ways by which the payment could be done that is for

2 or 7 dimes and 3 or 1 quarters.

1).Let us express 120 as the sum of two positive integers, one of which is divisible by 11 and the other by 17.

Then we can write 120 as :

$11x+17y=120$

Now, we will find the positive values of x and y .

$11(3)+17(-2)=-1$

Multiply both sides by (-120),we get

$11(3)(-120)+17(-2)(-120)=120$

Now,$11(17t)+17(-11t)=0$

So the general solution of the equation can be written as

$(17t-360,240-11t)$

The solution to be positive then

$17t-360\geq 0,240-11t\geq 0\\t\geq \frac{360}{17} ,t\leq \frac{240}{11} \\\frac{240}{11} \leq t\leq \frac{360}{17}$

Which is not an integer .

So we cannot express $120$ as the sum of two positive integers, one of which is divisible by $11$ and the other by $17.$

2) As 95 cents be paid for exactly using only dimes and quarters.

Therefore 95 can be  express as

$10x+25y=95$

The integral values of x and y satisfying the equation is

$x=2,y=3$ and $x=7,y=1$

$10.2+25.3=20+75\\=95$

And $10.7+25.1=70+25\\=95$

So there are two ways by which the payment could be done.

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