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# Problem on Positive Integers | PRMO-2019 | Problem 26 Try this beautiful problem from Algebra PRMO 2019 based on Positive Integers.

## Positive Integers | PRMO | Problem 26

Positive integers x,y,z satisfy xy+z=160 compute smallest possible value of x+yz.

• 24
• 50
• 29
• 34

### Key Concepts

Algebra

Integer

sum

PRMO-2019, Problem 26

Higher Algebra by Hall and Knight

## Try with Hints

First hint

x+yz=$\frac{160-z}{y}$+yz

=$\frac{160}{y}+\frac{z(y^{2}-1)}{y}=\frac{160-z}{y}+\frac{zy^{2}}{y}=\frac{160-z}{y}+zy$

for particular value of z, $x+yz \geq 2\sqrt{z(160-z)}$

or, least value=$2\sqrt{z(160-z)}$ but an integer also

Second Hint

for least value z is also

case I z=1, $x+yz=\frac{159}{y}+y$ or, min value at y=3 which is 56

case II z=2, $x+yz=\frac{158}{y}+2y$ or, min value at y =2 which is 83 (not taken)

case III z=3, $x+yz=\frac{157}{y}+3y$ or, min value at y=1 which is 160 (not taken)

case IV z=4, $x+yz=\frac{156}{y}+4y$ or, min at y=6 which is 50 (taken)

Final Step

case V z=5, $x+yz=\frac{155}{y}+5y$ or, minimum value at y=5 which is 56 (not taken)

case VI z=6, $x+yz=\frac{154}{y}+6y$ $\geq 2\sqrt{924}$>50

smallest possible value =50.

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Try this beautiful problem from Algebra PRMO 2019 based on Positive Integers.

## Positive Integers | PRMO | Problem 26

Positive integers x,y,z satisfy xy+z=160 compute smallest possible value of x+yz.

• 24
• 50
• 29
• 34

### Key Concepts

Algebra

Integer

sum

PRMO-2019, Problem 26

Higher Algebra by Hall and Knight

## Try with Hints

First hint

x+yz=$\frac{160-z}{y}$+yz

=$\frac{160}{y}+\frac{z(y^{2}-1)}{y}=\frac{160-z}{y}+\frac{zy^{2}}{y}=\frac{160-z}{y}+zy$

for particular value of z, $x+yz \geq 2\sqrt{z(160-z)}$

or, least value=$2\sqrt{z(160-z)}$ but an integer also

Second Hint

for least value z is also

case I z=1, $x+yz=\frac{159}{y}+y$ or, min value at y=3 which is 56

case II z=2, $x+yz=\frac{158}{y}+2y$ or, min value at y =2 which is 83 (not taken)

case III z=3, $x+yz=\frac{157}{y}+3y$ or, min value at y=1 which is 160 (not taken)

case IV z=4, $x+yz=\frac{156}{y}+4y$ or, min at y=6 which is 50 (taken)

Final Step

case V z=5, $x+yz=\frac{155}{y}+5y$ or, minimum value at y=5 which is 56 (not taken)

case VI z=6, $x+yz=\frac{154}{y}+6y$ $\geq 2\sqrt{924}$>50

smallest possible value =50.

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