LCM (remainders)

Question

A section of soldiers are rehearsing for the march past for the National Day parade . If they march in Pairs , one soldier will be without a partner . If they march in 3s , 5s, 7s , they will be a soldier short . Calculate the smallest possible number of soldiers in this section .

My workings -

Let X be the least number of soldiers in the section

If they march in 2s , 3s , 5s , 7s --> there will be one person in the section without a partner . So = $(n-1)$

And I took - $n-1 = LCM(2,3,5,7) $

$n-1 = 210 $

$ n = 211 $

However many said 211 is wrong and 209 is the answer . Can I get help? Thanks !!

Answer 1

Let $n$ be the number of soldiers. If you add one soldier, then the resulting number is divisible by $2,3,5,7$. Thus $n+1$ is divisible by $210$.

Remarks: $1$. The wording is suboptimal. The problem is that for "groups of $2$", they focused attention on the unpartnered soldier. For $3,5,7$ they say "will be a soldier short", which is fine. But the damage has been done, one misreads that as saying that a soldier will be without a partner, that there is an "extra" soldier. Note by the way that if one takes that interpretation, the correct answer is $1$, not $211$.

$2$. In the language of congruences, we have $n\equiv 1\pmod 2$, $n\equiv 2\pmod{3}$, $n\equiv 4\pmod{5}$, $n\equiv 6\pmod{7}$. This system of congruences is easier to handle if we rewrite it as $n\equiv -1\pmod{2}$, $n\equiv -1\pmod{3}$, $n\equiv -1\pmod{5}$, $n\equiv -1\pmod{7}$.