First, we will prove that this test works for divisibility by 19.

We can write the number we are testing as ,
where *y* is the units digit and *x* is the number represented by the
other digits.

Now, if this number is divisible by 19, then if we multiply it by 2, getting , it will still be divisible by 19.

If it is not, then 20x + 2y will not be either.

Then, if this number is divisible by 19, we can subtract 19*x*,
getting ,
and still have a number divisible by 19.

If this number is divisible by 19, our original number is also.

If it is not, neither is our original.

We can repeat this process until we get a number small enough to easily determine if it is divisible by 19 or not, such as a number less than or equal to 19.

Now, in the case of 29, we take the units digit of our original number, triple it and add it to the truncated number.

We can then repeat the process until we get a number less than or equal to 29.

If the number is 29, the original number is divisible by 29.

If the number is not 29, than neither is the original.

We are taking ,
(*y *is the units digit and *x *is the number represented by the
other digits) which will be divisible by 29 if and only if is.

will be divisible by 29 if and only if , our original number, is.

*Compute 1776 ^{1492!}(mod 2000) ; i.e., the
remainder when 1776^{1492!} is divided by 2000. (As usual, the exclamation point denotes
factorial.)*

Doing all calculations in (mod 2000), we get

1776^{1} ≡ 1776

1776^{2} ≡ 3154176 ≡ 176

1776^{3} ≡ 1776 176 ≡ 312576 ≡ 576

1776^{4} ≡ 1776 576 ≡ 1022976 ≡ 976

1776^{5} ≡ 1776 976 ≡ 1733376 ≡ 1376

1776^{6} ≡ 1776 1376 ≡ 2443776 ≡ 1776

…

We see that the pattern repeats with a period of 5.

We can easily show that 1376^{2} ≡ 1893376
≡ 1376
(mod 2000)

and thus 1376^{3} ≡ 1376 (mod 2000)

or 1376^{n} ≡ 1376 (mod 2000)

This means that if I take a power of 1776 that is divisible by 5, the result will be 1376 (mod 2000).

So, since 1492! ≡ 1
2 3 4 ** 5** …, it is a multiple of 5.

Therefore, 1776^{1492!} ≡ **1376 **(mod 2000)

The answer is **1376.**

** **

[Note: One can see that 1376 ≡ 0 (mod 16) and 1 (mod 125), so any power of this n would remain 0 (mod 16) and 1 (mod 125).]

**USAMTS Problem ****3/2/12**** **

*Given the arithmetic progression of integers,*

*308, 973, 1638,
2303, 2968, 3633, 4298,*

*determine the unique geometric progression of
integers,*

*b _{1},
b_{2}, b_{3}, b_{4}, b_{5}, b_{6}*

*so that*

*308 < b _{1}
< 973 < b_{2} < 1638 < b_{3} < 2303 < b_{4}
< 2968 < b_{5} < 3633 < b_{6} < 4298 .*

* *

* *

Given:

(Because it is a geometric progression.)

Let , where *p* and *q* are positive
integers and are prime to each other.

Now, , so

Since is divisible by *q*^{5},
*b*_{1 }_{ }*p*^{5} is also divisible by *q*^{5}.

Since *q* and *p* have
no common factors, *b*_{1} is divisible by *q*^{5}

^{ }

So,

*b*_{1} = *aq*^{5 }(where *a* is an integer, and *a*
__> __1).

Now our numbers *b*_{1},
*b*_{2}, … , *b*_{6} can be written as

*aq*^{5}, *apq*^{4}, *ap*^{2}*q*^{3}, *ap*^{3}*q*^{2},
*ap*^{4}*q*, *ap*^{5}

Since 4^{5} = 1024

and *aq*^{5} < 973

=> *q* < 4

or
*q* __<__ 3 *(1)*

Also, > 1 so *p* > *q* *(2)*

Also, = *r*^{4} __<__ (4297/974)

__>__ (3634/1637)

So *r* __<__
(4297/974)^{1/4} < 1.45

and *r* __>__ (3634/1637)^{1/4} >
1.22

So 1.22 < < 1.45

So if *q* = 3, then *p* = 4

If *q* = 2, then there are no possible values for *p*

The only possible values are *p* =4, *q *= 3 and *r* =

*b*_{1} = *a* 3^{5} = 243*a*

So 308 < 243*a* < 973

1.27 < *a* < 4.01

or 2 __<__ *a*
__<__ 4 --*(3)*

Also, *ap*^{5} > 3633

*a* 4^{5} > 3633

*a* > 3.5

Thus, this and *(3)* give *a* = 4

Thus,

*b*_{1} = 972, *b*_{2}
= 1296, *b*_{3} = 1728, *b*_{4} = 2304, *b*_{5}
= 3072, *b*_{6} = 4096

*Prove that every polyhedron has two vertices at which
the same number of edges meet.*

Let the polyhedron have
*n * vertices.

Now, the minimum number of edges that can meet at a vertex is 3.

The maximum number of edges that can meet at a vertex
is *n* 1

Now let us assume that we have a pigeon sitting on each
vertex of our polyhedron (We assume that these pigeons can count and read.),
and we have pigeonholes marked 3, 4, 5, … , *n* 1.

The pigeons count the number of edges that meet on their vertex, and fly to the pigeonhole with the same number.

Now, since we have *n * pigeons and only *n* 3
pigeonholes, there must be at least one pigeonhole with more than one
occupant.

The vertices from where these pigeon roommates originally flew from have the same number of edges meeting,

Thus, there are at least two vertices that have the same number of edges meeting.

(The argument is still valid even if we do not have pigeons that can count and read <smile>!)

* *

*to sides AB, BC, and CA, respectively, and intersect *

*at the points X, Y, and Z, as shown in the figure*

*on the right.
Determine the area of ΔABC if*

*each of the segments PQ, RS, and TU ***1**

*bisects (halves) the area of ΔABC, and*

*if the area of ΔXYZ is one unit. Your *

*answer should be in the form*

*,
where *a* and *b* are positive*

*integers.*

* *

Let AB = *c*, AC = *b*, and XY = *z*.

First, we will prove that .

since

and is common to both triangles

So,

Now, since and since

Also, since and

Thus,

The area of two similar triangles is proportion to the square of the ratio of the sides.

(This is proven on p. 556
of *Modern School Mathematics: GEOMETRY* by Jurgensen, Donnelly,
and Dolciani.)

It is given that the area of ΔABC is twice the area of ΔPQC, so PC = .

Since ΔABC ~ ΔPQC, we can set up a proportion to find PQ in
terms of *c*: .

Quadrilateral AUXP is a parallelogram, so AU = PX = .

Now, we can find *z*:

.

Now, .

*W* =

So, the area *W *of ΔABC
is .