Hey Kevin,

Good questions!

On that first one, I'd look at it this way: They're testing a divisibility number property, and usually the best way to look at those number properties is to get an idea of what the numbers will look like by testing small numbers. There's not much harm in listing out the first few instances of n for either case so that you can get a feel for the pattern:

When n/5 has a remainder of 1, your possibilities are:

1
6
11
16
21
26
31
(and any other positive integers ending in 1 or 6)

When n/7 has a remainder of 3, your possibilities are:

3
10
17
24
31
(and basically any other combination of a field goal and some touchdowns!)

Because the question asks you for the smallest value, you're really looking for the first intersection of these two patterns, which is n = 31. Had it asked something differently (which is a possible value of n, for example), you'd want to extrapolate the patterns for another intersection, but this one makes it fairly efficient by just looking at the smallest.

However, the trick here (which definitely took me a second) is that they're asking for k, and not for n. We need n + k to be a multiple of 35. Well, if k is 4, then 31 + 4 = 35, so we know that 4 is a possible value of k.

Because they asked for the smallest possible k, if any other choices were smaller than 4, you might want to do some more work to confirm that 4 is the smallest and not just the first one that came up. From the lists above, we know that n has to end in either 1 or 6, and that a multiple of 35 must end in either 0 or 5. If we're adding k to either 1 or 6, it will always be at least 4 away from that next threshold of 0 or 5, so we can safely conclude that 4 is our smallest possible k.


(Author's note, completely unrelated: since this question asks for the smallest "positive k", bonus points for anyone who can tell me the name of a song or album released by the recording artist "Positive K")


On that second question, that brings up a pretty huge strategic point that I come back to all the time in my classes - USE FRACTIONS!!!

1 1/4 feet is the same as 5/4 feet, and 24 feet is the same as 96/4 feet. So we have 96 "quarter feet" total and want to divide up into segments of 5. 96/5 is 19 with a remainder of 1 that we can discard.

Brian@VeritasPrep wrote:Hey Kevin,



(Author's note, completely unrelated: since this question asks for the smallest "positive k", bonus points for anyone who can tell me the name of a song or album released by the recording artist "Positive K")

I Got a Man!

LOL

Good work, tarn151! I was hoping someone would pick up on that... From the album "The Skills Dat Pay Da Bills"!

Okay, I can't help myself!

https://www.youtube.com/watch?v=VvYIpa1Ulvw

"So when I blow up, don't try to kick it to me later..."

Can't we approach this problem like this?

When positive integer n is divided by 5 the remainder is 1. when n is divided by 7, the remainder is 3. what is the smallest positive k such that k+n is a multiple of 35?


n = 5*a + 1
n = 7*b + 3, where a and b are two different integers.

5a + 1 = 7b + 3

Therefore, 7b = 5a - 2. That means, any multiple of 7 having the unit digit of either 3 or 8.
This condition satisfies when 7b is 28 or 63 (under 100 or when the number b having the unit digit of 4 or 9, like 14, 19 etc.). The corresponding value of 5a is 30 and 65 and the corresponding value of n is 31 and 66.

Now to find the value of k.

k+n should be a multiple of 35 (35, 70, 105, etc). So k is 4 (I think k is always 4).

Thanks,

Ren