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 Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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