A man has $1,000,000 he wishes to divide up in his will. He wants to give each person named in his will an amount of money, in dollars, which is a power of 7 ($7^{0}=$1, $7^{1}=$7, $7^{2}=$49, $7^{3}=$343, …).

He does not want to give more than six people the same amount. How can he divide the money?

**Answer:**

Lets call

**the number of people he leaves $1 to,**

*a***the number of people he leaves $7 to,**

*b*** the number of people he leaves $49 to, etc. The solution equation is thus:**

*c*1,000,000 = a + 7b + 49c + 343d + 2401e + 16807f + 117469g + 822283h.

Next divide 1,000,000 by 7. If 1,000,000 is divisible by 7 we could divide the money without giving anyone only $1. However 1,000,000/7 = 142857 plus a remainder of 1. By giving one person $1 (a=1) we now have:

999,999 = 7b + 49c + 343d + 2401e + 16807f + 117469g + 822283h.

Next divide by 7:

142,857 = b + 7c + 49d + 343e + 2401f + 16807g + 117469h.

Dividing 142,857 by 7 we get 20,408 plus a remainder of 1. So let b=1, divide by 7, and we get:

20,408 = c + 7d + 49e + 343f + 2401g + 16807h .

Dividing 20,408 by 7 we get 2,915 plus a remainder of 3. So let c=3, divide by 7, and we get:

2915 = d + 7e + 49f + 343g + 2401h.

Dividing 2915 by 7 we get 416 plus a remainder of 3. So let d=3, divide by 7, and we get:

416 = e + 7f + 49g + 343h.

Dividing 416 by 7 we get 59 plus a remainder of 3. So let e=3, divide by 7, and we get:

59 = f + 7g + 49h.

Dividing 59 by 7 we get 8 plus a remainder of 3. So let f=3, divide by 7, and we get:

8 = g + 7h.

Obviously g=1 and h=1.

**He should give:**

- 1 person $1
- 1 person $7
- 3 people $49
- 3 people $343
- 3 people $2401
- 3 people $16807
- 1 person $117649
- 1 person $823543.

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