Math Comp Qs (1 Viewer)

[Lukybear]

This Question I dont get from 2008:

How many different positive numbers are equal to the product of two odd one-digit numbers?

Its supposedly to be done using algebra, so can any1 pleasese enlighten me? Working out and explanation are very beneficial.

 

[study-freak]

I think it is:
No of different numbers formed by multiplication of two different one-digit odd numbers
=5C1x4C1/2 since for e.g. (1,5) and (5,1) give the same number.
=10.

No of different numbers formed by multiplication of two same one-digit odd numbers
=5C1x1
=5.

Total=15 different numbers.

 

[kurt.physics]

I think it is:
No of different numbers formed by multiplication of two different one-digit odd numbers
=5C1x4C1/2 since for e.g. (1,5) and (5,1) give the same number.
=10.

No of different numbers formed by multiplication of two same one-digit odd numbers
=5C1x1
=5.

Total=15 different numbers.

Close, but it is actually 14 because 1 x 9 is the same as 3 x 3.

 

[study-freak]

hmm didn't see that lol

 

[youngminii]

Kurt is a genius.

 

[Lukybear]

Study-Freak: How did you get the 4 in 5x4/2??

EDIT: O Nvm

 

[Lukybear]

Question:

1) What is the smallest whole number which gives a square number when multiplied by 2008?

2) A number is less than 2008. It is odd, it leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 5. What is the sum of the digits of the largest such number.

 

[khorne]

1) I won't do it, but you could reduce 2008 to all it's prime roots, and then find which roots are not to the power of a multiple of 2 and then make them....

Not sure if this would work but...

Let me try actually:

so it's 2^3 x 251 (which is prime)
Therefore it must be multiplied by 2x251 = 502

502*2008 = 1008016 sqrt answer = 1004 So I think it's right

Did you guys know 251 is a http://en.wikipedia.org/wiki/Sexy_prime

 

[kurt.physics]

1) I won't do it, but you could reduce 2008 to all it's prime roots, and then find which roots are not to the power of a multiple of 2 and then make them....

Not sure if this would work but...

Let me try actually:

so it's 2^3 x 251 (which is prime)
Therefore it must be multiplied by 2x251 = 502

502*2008 = 1008016 sqrt answer = 1004 So I think it's right

Did you guys know 251 is a Sexy prime - Wikipedia, the free encyclopedia

thats what i got.

 

[kurt.physics]

Question:

1) What is the smallest whole number which gives a square number when multiplied by 2008?

2) A number is less than 2008. It is odd, it leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 5. What is the sum of the digits of the largest such number.

2)Let this number be m

m = 3n + 2
m = 5p + 4

m + 1 = 3(n + 1) (1)
m + 1 = 5(p + 1) (2)

Also, m is odd, and so when we add 1, it will be even, and so is divisible by 2.

So m+1 is a multiple of 3 from (1) and a multiple of 5 from (2) and also a multiple of 2. So m + 1 is also a multiple of 30.

So we can look for the largest multiple of 30 thats under 2008. This number is 1980. So

m+1 = 1980

.: m = 1979

Thus the sum is 1 + 9 + 7 + 9 = 26

 

[kurt.physics]

If you know modular arithmatic, then the solution is simple and elegant.

Letting this number be m,



also,



and as m is odd,





The largest number -1 mod 30 is 1979

Thus the sum and hence answer is:

 

[ninetypercent]

what geniuses.

 

[thetai903]

Hum...
I would not be suprise to see kurt.physics come 1st in the state for something. What school are you from, kurt?

 

[kurt.physics]

Hum...
I would not be suprise to see kurt.physics come 1st in the state for something. What school are you from, kurt?

Not a fancy one, im from Woolgoolga High School.

 

[youngminii]

I'm guessing you're going to be the first time your school gets a student that comes first in the state for Maths.
Damn you're like Namu x2.

 

[khorne]

Nice you see you're a psychic..=]

 

[Lukybear]

1) I won't do it, but you could reduce 2008 to all it's prime roots, and then find which roots are not to the power of a multiple of 2 and then make them....

Not sure if this would work but...

Let me try actually:

so it's 2^3 x 251 (which is prime)
Therefore it must be multiplied by 2x251 = 502

502*2008 = 1008016 sqrt answer = 1004 So I think it's right

Is there a speicific rule for this? And where did you learn such things?

 

[kurt.physics]

Is there a speicific rule for this? And where did you learn such things?

There are a couple of "rules". Firstly what khorne did was prime factorise the number, prime factorization is where you make a number completely out of primes, because every number can be expressed as the product of primes. To do so, you divide a number by a small prime and then continue this process until you end up at a prime number.

For example, take 72. The last digit is a multiple of 2, so we can easily tell that 2 is a factor. Dividing by 2 we get 36. So 72 = 2 x 36. But 36 is not a prime, so we observe that 36 is also divisible by 2. Dividing by 2 we get 18. So 36 = 2 x 18, and so 72 = 2 x (2 x 18) = 2 x 2 x 18. Continuing in this way we find 72 = 2 x 2 x 2 x 3 x 3

This can be written as 72 = 2³ x 3².

Why do we prime factorize? We do this because it reveals alot of information about that number.

Another rule is to understand that for a number to be a perfect square, the prime factorization MUST have each numbers exponent as a multiple of 2. For example, lets look at the number 72 again, recall its prime factorization is 2³ x 3²

Take the square root of 72



remember:



so,





Now,

can easily be found, it is 3, because:



but cannot be easily found, because as you know, it equals . So the square root of 72 is . But this number is not an integer, and so 72 is not a perfect square (as the root of a square must be an integer).

But if it were then the square root would be

So just note that a perfect squares prime factorization will have all its exponents (powers) as a multiple of 2.

The prime factorization of 2008 is 2³ x 251.

So to make this a perfect square, we must multiply the 2³ by 2 to get the power to be 4 (which is a multiple of 2), and to get 251 to be a multiple of 2 we must multiple 251 by 251 to get 251².

The new number is 2⁴ x 251<sup>2, and so as all its powers are a multiple of 2, then the number is a perfect square.

But the numbers we have to multiple 2008 by to get this perfect square by was 2 x 251, which is 502. This is indeed the answer.</sup>