Minlcm

Suppose that $a,b,c,d \in \mathbb{N}$ , with $a < b < c < d$ and $a+b+c+d=1000$ , and write $N = \mathrm{lcm}(a,b,c,d)$ . Then $1000 < 4d$ , and so $d > 250$ . If $N > d$ , then $N = kd$ for some $k > 1$ , and hence $N \ge 2d > 500$ . Thus values of $N \le 500$ are only obtained when $N = d$ .

Suppose now that $N \le 500$ , and that $N = d$ . Then $a,b,c$ all divide $d$ , and hence $a \le \tfrac14d$ , $b \le \tfrac13d$ , $c \le \tfrac12d$ , and hence $1000 \; = \; a + b + c + d \; \le \; \big(\tfrac14 + \tfrac13 + \tfrac12 + 1\big)d \; = \; \tfrac{25}{12}d$ so that $d \ge 480$ . Thus we deduce that $N \ge 480$ in all cases.

The case $a = 120$ , $b = 160$ , $c=240$ , $d=480$ shows that $N=480$ is possible. The smallest possible value of $N$ is $\boxed{480}$ .

Suppose that $\text{lcm}(a,b,c,d)=ka=lb=mc=nd$ where $k,l,m$ and $n$ are distinct. Rewriting $b,c$ and $d$ in terms of $a$ , gives the equation $a+\frac kl a+\frac km a+\frac kn a=1000$ This rearranges to $\large \text{lcm}(a,b,c,d)=ka = \frac{1000}{\frac 1k + \frac 1l + \frac 1m + \frac 1n}$ We can see that this is minimized when $k=1, l=2, m=3, n=4$ giving the answer $\boxed{480}$

If $a$ is the largest number of the four, it will be over $250$ . This means any $\text{lcm}$ greater than $a$ will be significantly greater than $a$ . Best case scenario, we want $\text{lcm}(a,b,c,d)=a$ .

To minimise $a$ , we need to make $b$ , $c$ & $d$ as large as possible, so we make these the largest distinct fractions of $a$ : $\frac{a}{2}$ , $\frac{a}{3}$ & $\frac{a}{4}$ . Now we check to see if this gives us integer solutions for $a$ , $b$ , $c$ & $d$ :

$a+\frac{a}{2}+\frac{a}{3}+\frac{a}{4}=1000$ $a\times\frac{25}{12}=1000$ $a=480$ $b=\frac{480}{2}=240$ $c=\frac{480}{3}=160$ $d=\frac{480}{4}=120$

As these are all integer solutions, we know that the minimum possible $\text{lcm}$ is $\boxed{480}$ .

If a, b, c, d are integers and if d < c < b < a and if a+b+c+d = p, a given number,

lcm (a,b,c,d) has the lowest possible value if b = 1/2 a, c = 1/3 a and d = 1/4 a, and then this value (for lcm) equals a.

If p = 1000, a + 1/2a + 1/3a + 1/4a = a + 13/12 a = 25/12 a = 1000, hence a = 12 x 1000/25 = 12 x 40 = 480, hence lcm (a,b,c,d) = 480.