I just found a number with and interesting property:?

Let's use congruence notation.

We have to solve the 9 congruences

x = 1(mod 2)

x = 2(mod 3)

x = 3(mod 4)

x = 4(mod 5)

x = 5(mod 6)

x = 6(mod 7)

x = 7(mod 8)

x = 8(mod 9)

x = 9(mod 10)

But we can eliminate the first 4 of these right away.

Nos. 1 and 3 are included in no.7

and no. 4 is included in no. 9

Let's solve nos 5,7,8 and 9. Then we'll put that together with no.6

to get the final answer.

5 and 7:

x = 5(mod 6)

x = 7(mod 8)

Change both to their LCM = 24

4x = 20(mod 24)

3x = 21(mod 24)

x = -1 = 23(mod 24)

Next, combine the solution with no.8. LCM = 72

x = 23(mod 24)

Note that any number congruent to 23(mod 24) is

congruent to 23, 47 or 71(mod 72).

Only 71 = 8(mod 9).

Next step: Combine this result with no. 9

x = 71(mod 72)

x = 9(mod 10)

LCM is 360.

Let's use the idea of the previous step:

If x = 71(mod 72) then

x = 71, 143, 215, 287 or 359(mod 360)

Only 359 = 9(mod 10)

Now for the final step:

x = 359(mod 360)

x = 6(mod 7)

LCM = 2520.

We can write these, more simply, as

x = -1(mod 360)

x = -1(mod 7)

7x = -7(mod 2520)

360x = -360(mod 2520)

353x = -353(mod 2520)

x = -1 = 2519(mod 2520).

So the smallest positive number that satisfies all the

congruences is x = 2519.

All solutions are therefore of the form

x = 2519 + 2520k,

where k is any positive integer.

Whew!

That was the hard solution.

I just found a much simpler one!

Let x be the smallest solution.

The conditions of the problem require that

x+1 be equal to the LCM of 2,3,4,5,6,7,8,9 and 10.

Why?

Well, we can write each of the 9 congruences as

x = -1(mod 2)

x = -1(mod 3)

x = -1(mod 4)

...................

x = -1(mod 10)

so x+1 will be divisible by all these numbers

and so it will be divisible by their least common

multiple.

As pointed out above, it is enough to

find the LCM of 6,7, 8, 9 and 10.

The easiest way to do this is to do them 2 at a time.

The LCM of 2 numbers is their product divided by

their GCF

So LCM(6,8) = 6*8/2 = 24

LCM(24,9) = 24*9/3 = 72

LCM(72,10) = 72*10/2 = 360

LCM(360,7) = 360*7 = 2520.

So x+1 = 2520

and x = 2519, as before.

Let N be your number. Notice that if you add 1 to your number

then the result, N + 1 , is divisible by 2 through 9 since the

remainders all become zero. The smallest such number

divisible by 2 through 9 is the least common multiple of

2 through 9 which is 2520 = N + 1. Then, N = 2519.

2519

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