Today's Problems

40 1. What is the smallest number divisible by both 8 and 10?
A lot of you did not read the question carefully and put down 2.

9/25 2. Which fraction is a ratio of relatively prime numbers? 15/60 9/25 8/144 49/14
This means which is reduced to lowest form? Clearly 15 and 60 are both divisible by 5, 8 and 144 are both divisible by 2 and 49 and 14 are both divisible by 7.

33 3. What is the smallest integer, n, such that 1n1 is a perfect cube?
This one is a bit tricky, but thinking clearly will make it easy. 8 is a perfect cube because it is 2x2x2. 27 is a perfect cube because it is 3x3x3. So what number times itself twice more has a product starts with a 1 and ends with a 1? Well it can't be an even number. 1x1x1=1 so it could end in a 1. 3x3x3 =27 so that will end in a 7, not a 1. 5x5x5 nope - that'll end in a 5. 7x7x7=49x7 Nope-that'll end in a 3. 9x9x9=81x9 which will end in a 9. So it has to end in a 1! Let's try 11x11x11. Multiply it out and you get 1331 which fits the pattern we need! So the answer is 11. Wait! What was the question? 1n1 is a perfect cube - ah 1331 is the perfect cube so n is 33.

18,900 and 36,450 4. Find two pairs, of integers that have a Greatest Common Factor of 18 and a Least Common Multiple of 900. So we're looking for two groups of factors with a GCF of 18 and a LCM of 900.
The factors of 18 are 2 x 3 x 3. The factors of 900 are 3 x 3 x 5 x 2 x 5 x 2
So we have two groups of factors - 2,3,3 and 2,2,3,3,5,5
Both have factors 2 x 2 x 3 so both have a greatest common factor of 18. So one pair is: 18 and 900.
To get the other pair, lets move one of the factors of 900 into the 18 camp - but we can't move a 5, because then(2,3,3,5 2,2,3,3,5) the greatest common factor wouldn't be 18 any longer - it would be 18x5 which is 90. So what else can we move? We need a 2,3,3 in both our numbers, so 2 is the only factor left. So let's shift a 2 from the 900 to the 18: 2,2,3,3 and 2,3,3,5,5
2x2x3x3=36 and 2x3x3x5x5 =450 Our new pair with with a GCF of 18 and a LCM of 900 is 36 and 450.

(5x+6)/15 5. Write as on fraction in reduced form: x/3 + 2/5 = ?
How do add fractions? Get a common denominator. The common denominator is 15. Multiply top and bottom of x/3 by 5 to get 5x/15 and multiply top and bottom of 2/5 by 3 to get 6/15.
5x/15 + 6/15 can be now added because they have a common denominator. Answer (5x+6)/15