12

With some guess work I found the answers to be 8 and 12.

But is there any general formula for this?

Note that the question is asked to my nephew who is at 4th grade.

Tanmoy
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    No, but it's certainly possible to write down an algorithm. – Simon S Feb 11 '15 at 16:03
  • I would guess that if the question was asked to a 4th-grade student, then the intention was for him to find the answer using trial & error, rather than developing a general-case formula for this. Nevertheless, it should be interesting to see if such formula exists (which, BTW, @SimonS: to the best of my knowledge, every deterministic algorithm can be converted to a formula). – barak manos Feb 11 '15 at 16:08
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    @barakmanos Only if you know when the algorithm will terminate. Try to come up with a general formula for Newton's method for example ;) – AlexR Feb 11 '15 at 16:10
  • @AlexR: I might give it a try, but it should be interesting to post it as a question (I'm willing to bet that someone here will come up with an answer faster than you think). – barak manos Feb 11 '15 at 16:11
  • every deterministic algorithm can be converted to a formula. Hm, yes, but writing down a formula for even this problem would be fairly daunting. Delighted if you can prove me wrong. – Simon S Feb 11 '15 at 16:12
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    Depending on the conventions adopted for an LCM involving negative numbers, -4 and 24 could be another solution. – Peter Taylor Feb 12 '15 at 12:32

7 Answers7

20

With 4th grade knowledge in mind, I would say this question is about finding all the factors of $24$, because the two numbers must both be factors of $24$ or they would have a different least common multiple.

So listing the factors of $24$ we get $1,2,3,4,6,8,12,24$. Then it's easy to see which two add to $20$, although you could certainly use the exercise, if so inclined, to develop a few more rules for harder cases, and in another case you might need to be a little careful that the least common multiple is indeed still $24$.

Another approach: $\frac{20}{24} = \frac{5}{6}$ - can we write $\frac{5}{6}$ as the sum of two unit fractions? The answer is yes, $\frac{1}{2} + \frac{1}{3}$, and the numbers we seek are $\frac{24}{2}=12$ and $\frac{24}{3}=8$

Joffan
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  • This is I find the easiest way to explain it :). Thanks – Tanmoy Feb 11 '15 at 16:42
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    @Tanmoy you're welcome; I added a link to Egyptian fractions for the alternative approach, might be of interest for your nephew. – Joffan Feb 11 '15 at 16:43
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    Except that in general, picking two factors of 24 doesn't mean that their LCM is 24. – Hao Ye Feb 12 '15 at 01:42
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    @HaoYe which in fact I say at the end of the second paragraph. The point is that picking numbers other than the factors of $24$ will guarantee that the LCM is not 24. – Joffan Feb 12 '15 at 02:42
  • My bad - must have missed that part of the sentence. – Hao Ye Feb 12 '15 at 02:53
12

A general algorithm

If $a+b = n$ and $\mathrm{lcm}(a,b) = c$ we let $\gcd(a,b) = d$ to get $$ab = cd\\ n = a+b$$ Solving $b = n-a$ gives us $$a(n-a) = cd \\ \Leftrightarrow a^2-na + cd = 0 \\ \Leftrightarrow a = \frac n2 \pm \sqrt{\frac{n^2}4 - cd}$$ So for even $n$ we must find $d$ such that $\frac{n^2}4 - cd$ is a perfect square (since $c>0$ this will amount to a finite number of possibilities). The $\pm$ is irrelevant because $b$ will take the alternate value.
If $n$ is odd, $n^2 - 4cd$ must be a perfect square instead and we obtain an analogous formula: $$a = \frac12 (n \pm \sqrt{n^2 - 4cd})$$

In both cases, $d$ can range between $1$ and $\left\lfloor \frac{n^2}{4c} \right\rfloor$

Your case thus allows $1\le d\le \lfloor\frac{400}{96}\rfloor = 4$ and $100-24d$ must be a perfect square. $d=4$ yields $4 = 2^2$ so $$a = 10 + \sqrt{4} = 12; \quad b = 10 - \sqrt4 = 8$$

AlexR
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7

With an LCM of 24 one of the numbers must be 8, as $2^3$ divides 24. It can't be 16 as 16 does not divide 24.

JMP
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  • Why can't the number be less than 2^3? Like 4 or even 2? – Tanmoy Feb 11 '15 at 16:17
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    @Tanmoy: Because the LCM would be smaller. One of the numbers needs to contribute a factor $2^3$ – Ross Millikan Feb 11 '15 at 16:20
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    Note that the argument is only correct because $24 = 2^3\cdot 3$ so if $8$ was not one number, one had to be $24 > 20$ which contradicts that their sum is $20$. – AlexR Feb 11 '15 at 16:35
  • @AlexR, that depends on what you consider to be the value of $\textrm{lcm}(-4, 24)$. – Peter Taylor Feb 12 '15 at 11:55
  • @PeterTaylor Why would it be any different from $24$? If we allow integer arguments, we must allow integer multipliers and define $\mathrm{lcm}:\mathbb Z^2 \to \mathbb N$ for uniqueness. – AlexR Feb 12 '15 at 12:01
  • @AlexR, it's a question of conventions, and I'm sure there are some people who would argue that an lcm involving a negative number should be negative. But since you say it's 24, that's a counterexample to your earlier claim. – Peter Taylor Feb 12 '15 at 12:24
  • @PeterTaylor Interesting, but my earlier claim is only about the lcm of natural numbers, which is not ambiguous and appropriate in the context of the question at hand. – AlexR Feb 12 '15 at 12:29
  • Makes sense. The LCM of (2^a3^b5^c, 2^x3^y5^z) is 2^max(a,x) * 3^max(b,y) * 5^max(c,z). For LCM=24 we know that max(a,x)=3, max(b,y) is 1, and max(c,z)=0. Thus we need to check the possibilities {a,b} = {3,1} (fails, 24>20) and {a,b} = {3,0} (the conclusion above) – MSalters Feb 12 '15 at 13:05
6

You want two factors of $24$ that add up to $20$. That means one factor must be less than or equal to $10$ (i.e. $20/2$), and the other must be between $10 $ and $20$ (inclusive) since it is equal to the first subtracted from $20$.

The only factor of $24$ between $10$ and $20$ inclusive is $12$. $20-12 = 8$. $8$ is a factor.

So the solution is $8$ and $12$.

user103828
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Rob
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5

Let $x$ and $y$ be the numbers. Note that $xy$ is the product of the LMC and the gcd, say, $d$. Then: $$xy=24d$$ $$x+y=20$$

I don't know if at 4th grade the pupils are supposed to be able of solve systems. If it is the case, you can write $$y=20-x$$ and then $$x(20-x)=24d$$

Solve $x$ to get $$x=\frac{20\pm\sqrt{400-96d}}2=10\pm2\sqrt{25-6d}$$

Therefore, $25-6d$ must be a perfect square. This only happens if $d=4$, which yields $x=8$ or $x=12$.

This is not a formula, but it makes the search much easier.

ajotatxe
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3

Let the numbers be $a,b$ and let $\gcd=d$,

then $a=pd, b=qd$

$a+b=d(p+q)=20$

also $a.b=(l.c.m)(\gcd) $

or, $pqd^2=(l.c.m)(d)$, or, $l.c.m=pqd=24$

dividing the two equations we get,

$\frac{p+q}{pq}=\frac{20}{24}=\frac{5}{6}$

or,$6(p+q)=5pq$

$5$ does not divide $6$, therefore, $5$ divides$(p+q)$, also, $p+q$ does not divide pq, or, $p+q$ divides $5$, $\implies$ $p+q=5$ and $pq =6$ and $d=4$

solve for $p$ and $q$ to get $a$ and $b$

Shobhit
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0

Let the numbers be $a, b$.

$\gcd(a,b) = \gcd((a+b), lcm(a,b))$ (The reason is explained later)

The $lcm(a,b)$ is given. Thus, you can find the product of the numbers using the formula: $ab=lcm(a,b)*gcd(a,b)$.

$(a+b)$ is also given. Now, you can easily solve these 2 equations.

For this particular question,

$a+b=20$ ----------(1)

$gcf(20,24)=4$

$ab=24*4=96$ ----------(2)

Let the numbers be $a,b$ and their $\gcd=d$,

Then $a=pd, b=qd$, where $p,q$ are co-prime.

$a+b=d(p+q)$

$lcm(a,b)=d(pq)$

Now, because $p, q$ are co-primes, $p+q$ and $pq$ will be co-prime too. i.e., $gcd(p+q,pq)=1$. Visit this question for the explanation.

Thus, $\gcd(d(p+q),d(pq))=d$

$\implies$ $\gcd((a+b),lcm(a,b))=d=\gcd(a,b)$

I am not experienced in writing mathematics symbol markdowns. I tried my best. Please edit if required.

Sourav Ghosh
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