1) We note here that each specific divisor x (3, 4, 5, 6 or 7) of the desired integer N yields a remainder of (x - 2). Therefore, each specified divisor x of N + 2 yields a remainder of 0. In other words, N +2 is a multiple of 3, 4, 5, 6 and 7. The smallest such number is:

420 (2 **·** 2 **·** 3 **·** 5 **·** 7)

If N + 2 = 420 then the desired number N is: N = 420 - 2 = 418

Following on from this:

418/ 3 = 139 r1

418/4 = 104 r2

418/5 = 83 r3

418/6 = 69 r4

418/7 = 59 r5

2) Write the formula for the volume of a cone:

V = 1/3 (pr^{2} h)

Also, for this situation: r^{2 } + h ^{2} = 1: V= 1/3 pr^{2} **Ö**(1 - r^{2})

We see the volume V is zero when r = 0 or 1. Hence, the maximum ratio must be found for some r between 0 and 1. How to obtain it? We first introduce the formula for the surface area of a cone:

A = pr h _{s } where h _{s } denotes the *slant height* of the cone for which:

h _{s }= **Ö **(r^{2 } + h ^{2})

Now, because in our case: h _{s }= 1 then A = pr.

Then the ratio V/ A = 1/3 pr^{2} **Ö**(1 - r^{2})/ pr

Or, more simply:

V/ A = 1/3 r **Ö**(1 - r^{2})

Differentiation of a function must be used at this point. The maximum or minimum of a differentiable function occurs at critical points where the derivative is zero. Then let (V/A) be a differentiable function, e.g.

(V/ A)' = 1/3 r **Ö**(1 - r^{2})

By the product rule:

1/3 r **Ö**(1 - r^{2}) = 1/3 **Ö**(1 - r^{2}) + 1/3 r **Ö**(1 - r^{2})

Using the chain rule for derivatives:

(V/ A)' = 1/3 **Ö**(1 - r^{2}) + 1/3 r (½) **Ö**(1 - r^{2}) (2r)

= 1/3 **Ö**(1 - r^{2}) - r^{2}/3 **Ö**(1 - r^{2})

We then set the derivative to 0 and solve for r:

1/3 **Ö**(1 - r^{2}) - r^{2}/3 **Ö**(1 - r^{2}) = 0

Multiply both sides by: 3 **Ö**(1 - r^{2}):

0= (1 - r^{2}) - r^{2} = 1 - 2r^{2}

Or: 2r^{2 }= 1

è

r = **Ö**1/2 = **Ö2**/2

Then Max (V/A) = (1/3) (**Ö2**/2) (**Ö2**/2) = 2/ 12 = 1/6

Thus the maximum ratio (V/A) occurs when:

r = **Ö**2/2 and h = **Ö**2/2

This can be achieved when one full quadrant of a circle (sector angle 90 deg) is removed from a paper circle and the straight edges as re-connected.

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