Which of the following is the lowest positive integer...

[RonPurewal]

A student posed the following question, which I'm posting on the forums for maximum benefit:

Which of the following is the lowest positive integer that is divisible by 8, 9, 10, 11, and 12?


7,920

5,940

3,960

2,970





Here I immediately discard the first option because in the probl text there is the constraint that says 'the lowest positive integer'. Since the left solution options are surely multiple of 8,9,10 and 12, I need to check if they multiple of 11 as well. I start with the thid option (C) and it turns out to be the number divisible for 8,9,10,11,and 12. So I pick answer choice C, which is the correct one. I'd like to know from you if the move of discarding right away the first option solely based on the constraint given was a smart one or a coincidence.

[RonPurewal]

---

It's a coincidence.

"The smallest number that does xxxx" is just some number. If you have five choices, that number could be any of the five choices.

If you just take the same idea that's in this problem, but make it simpler, you'll see what I mean.

What is the smallest positive integer that's divisible by both 8 and 9?
(a) 24 (b) 32 (c) 36 (d) 48 (e) 72

The answer to this question is 72, and 72 is the biggest choice here.

You could just as easily make 72 the second-greatest, third-greatest, fourth-greatest, or smallest value.

[RonPurewal]

Also, if the question asks for the smallest value that does something, you shouldn't start in the middle of the choices. If you want the SMALLEST possible value, you should start with the SMALLEST answer choice, and work your way up until you get an answer choice that works.

If you do that, then, as soon as you get a choice that works, you are guaranteed that it's the smallest one (since you've been working from smallest to biggest).

If you start in the middle of the choices, then, even if the middle choice works, it's still possible that one of the smaller choices might work, too.

E.g., consider the simpler problem that I posted above, but this time with answer choices 36, 72, 144, 216, and 360.
If you start with 144, then it "works""”i.e., it's divisible by both 8 and 9"”but it's not the right answer, because there's a smaller choice (72) that also works.
If you start with 36 (which doesn't work"”it's not divisible by 8) and then work up to 72, you'll be guaranteed that 72 is the smallest number that works.

[RonPurewal]

I'm also curious about this:

Since the left solution options are surely multiple of 8,9,10 and 12,

When you say "surely a multiple of...", it seems as though you're saying this is obvious"”i.e., that you can just tell from a glance at the numbers, without having to do any arithmetic.

Is that what you're saying?
If so, I'm curious what method/criterion you're using to determine whether the numbers are multiples of 8 and 12.
I know decently easy divisibility tests for 9 (and divisibility by 10 is trivial), but the only way I could judge divisibility by 8 and/or 12 would be to divide the numbers by 8 and/or 12.

[georgepa]

It might be simplest to test divisibility of the not obvious numbers.


However, there are some shortcuts for divisibility like Ron mentioned

A number is divisible by

• 8 - if the last 3 digits are divisible by 8


• 9 - if the sum of the digits are divisible by 9


• 10 - if the last digit is zero


• 11 - if the difference of A and B is 0 or divisible by 11 where
  
  
  
  • A = sum of every 2nd digit
  
  
  • B = sum of other digits
  
  
  • 64614 => (4+1) -(6+6+4) = 5 - 16 = -11 (-11 is divisible by 11)
  
  
  • 6347 => (3 + 7) - (6 + 4) = 0


• 12 - if the number is divisible by 3 and 4


• 3 - if the sum of the digits is divisible by 3


• 4 - if the last two digits are divisible by 4


• 5 - if the last digit is a 5 or 0


• 6 - if the number is even and divisible by 3


• 7 - there is a shortcut - but its time consuming - so you should simply do long division

[georgepa]

a) 7,920

b) 5,940

c) 3,960

d) 2,970

Applying shortcuts above you can eliminate everything except 8. Test divisibility for 8 from smallest to highest

d) 2,970 => 970/8 = not int
c) 3,960 => 960/8 = 120 => divisible - is the smallest that works

[RonPurewal]

Thanks for the list above.

I would recommend those divisibility tests only if at least one of the following three things is true:
(a) you are very, very good at memorizing things;
(b) you can intuitively understand the test; or
(c) you happen to be extremely slow at long division.

As an example of (b), I know that 1000 is divisible by 8, so that's why I can figure out that it's ok to ignore everything except the last three digits of a number.

If (c) is true, then you should probably just practice long division for a bit.

In any case, these tests should not be very high on anyone's list of priorities, simply because the time savings involved is so small. I.e., it's possible that these tests will save time compared to long division, BUT
1/ if there are such time savings involved, then the savings will be on the order of a few seconds at the very most;
2/ some of these tests -- in particular the one for 11 -- seem much harder than just doing long division (especially since multiples of 11 are such easy numbers).

If you're spending more than two minutes on these divisibility tests, then there's definitely something more important you could be doing instead.