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## Ratio of LCM & GCF | Algebra | AMC 8, 2013 | Problem 10

Try this beautiful problem from Algebra based on the ratio of LCM & GCF from AMC-8, 2013.

## Ratio of LCM & GCF | AMC-8, 2013 | Problem 10

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?

• 310
• 330
• 360

### Key Concepts

Algebra

Ratio

LCM & GCF

Answer:$330$

AMC-8, 2013 problem 10

Challenges and Thrills in Pre College Mathematics

## Try with Hints

We have to find out the ratio of least common multiple and greatest common factor of 180 and 594. So at first, we have to find out prime factors of 180 & 594. Now…….

$180=3^2\times 5 \times 2^2$

$594=3^3 \times 11 \times 2$

Can you now finish the problem ……….

Now lcm of two numbers i.e multiplications of the greatest power of all the numbers

Therefore LCM of 180 & 594=$3^3\times2^2 \times 11 \times 5$=$5940$

For the GCF of 180 and 594, multiplications of the least power of all of the numbers i.e $3^2\times 2$=$18$

can you finish the problem……..

Therefore the ratio of Lcm & gcf of 180 and 594 =$\frac{5940}{18}$=$330$

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## Unit digit | Algebra | AMC 8, 2014 | Problem 22

Try this beautiful problem from Algebra about unit digit from AMC-8, 2014.

## Unit digit | AMC-8, 2014|Problem 22

A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the unit digit of the number?

• 7
• 9
• 5

### Key Concepts

Algebra

Multiplication

integer

Answer:$9$

AMC-8, 2014 problem 22

Challenges and Thrills in Pre College Mathematics

## Try with Hints

Let the ones digit place be y and ten’s place be x

Therefore the number be $10x+y$

Can you now finish the problem ……….

Given that the product of the digits plus the sum of the digits is equal to the number

can you finish the problem……..

Let the ones digit place be y and ten’s place be x

Therefore the number be $10x+y$

Now the product of the digits=$xy$

Given that the product of the digits plus the sum of the digits is equal to the number

Therefore $10x+y=(x\times y)+(x+y)$

$\Rightarrow 9x=xy$

$\Rightarrow y=9$

Therefore the unit digit =$y$=9