Find the smallest number of 5 digits exactly divisible by 16,24,36,54
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What's the source of this question? It seems remarkably un-GMATesque, since you need to do a lot of crunching to come up with the right answer; especially given the inclusion of choice E, which makes backsolving almost impossible.
That said, let's do some crunching. When you see big numbers and factors or multiples, your first instinct should be to break things down into primes.
Let's break down our component numbers and create the lowest common multiple:
16 = 2*2*2*2
so, our LCM must have 2^4 in it
24 = 2*2*2*3
we already have lots of 2s, so we just need to bring in the 3. Our work in progress is now:
2^4 * 3
36 = 4*9 = 2*2*3*3
we already have lots of 2s and one 3, so we need to bring in one more 3. Our current LCM:
2^4 * 3^2
54 = 2*27 = 2*3*3*3
we need to add 1 more 3, giving us a final LCM of:
2^4 * 3^3
The question asks what's the smallest 5 digit number that's a multiple of our LCM. Here's where the question gets unfair, since there's no elegant (i.e. short and sweet) solution - brute force is now required, something that almost never happens on the GMAT.
Well, our number is 16*27 = 432
432 * 20 = 8640... that's a good starting point, let's work up from there:
8640 + 432 = 9072 + 432 = 9504 + 432 = 9936 + 432 = 10368... choose D.
So, not only is this a poorly constructed question, but (assuming that you copied it correctly) the answer provided is also wrong!
That said, let's do some crunching. When you see big numbers and factors or multiples, your first instinct should be to break things down into primes.
Let's break down our component numbers and create the lowest common multiple:
16 = 2*2*2*2
so, our LCM must have 2^4 in it
24 = 2*2*2*3
we already have lots of 2s, so we just need to bring in the 3. Our work in progress is now:
2^4 * 3
36 = 4*9 = 2*2*3*3
we already have lots of 2s and one 3, so we need to bring in one more 3. Our current LCM:
2^4 * 3^2
54 = 2*27 = 2*3*3*3
we need to add 1 more 3, giving us a final LCM of:
2^4 * 3^3
The question asks what's the smallest 5 digit number that's a multiple of our LCM. Here's where the question gets unfair, since there's no elegant (i.e. short and sweet) solution - brute force is now required, something that almost never happens on the GMAT.
Well, our number is 16*27 = 432
432 * 20 = 8640... that's a good starting point, let's work up from there:
8640 + 432 = 9072 + 432 = 9504 + 432 = 9936 + 432 = 10368... choose D.
So, not only is this a poorly constructed question, but (assuming that you copied it correctly) the answer provided is also wrong!
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