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2002 AIME II Problems/Problem 7: Difference between revisions

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<math>k \equiv 24 \pmod{25}</math>
<math>k \equiv 24 \pmod{25}</math>
<math>k \equiv 15 \pmod(16)</math>
<math>k \equiv 15 \pmod{16}</math>


is one pairing, and
is one pairing, and
Line 40: Line 40:
The chain is as follows:
The chain is as follows:


<math> 16 \pmod{25}</math>
<math>16 \pmod{25}</math>
7, 23, 14, 5, 21, 12, 3, 19, 10, 1, 17, 8, 24, 15, 6, 22, 13, 4, 20, 11, 27, 18, 9, 0, and then it loops.
7, 23, 14, 5, 21, 12, 3, 19, 10, 1, 17, 8, 24, 15, 6, 22, 13, 4, 20, 11, 27, 18, 9, 0, and then it loops.


Revision as of 18:35, 23 September 2017

Problem

It is known that, for all positive integers $k$,

$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.

Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.

Solution

$\frac{k(k+1)(2k+1)}{6}$ is a multiple of $200$ if $k(k+1)(2k+1)$ is a multiple of $1200 = 2^4 \cdot 3 \cdot 5^2$. So $16,3,25|k(k+1)(2k+1)$.

Since $2k+1$ is always odd, and only one of $k$ and $k+1$ is even, either $k, k+1 \equiv 0 \pmod{16}$.

Thus, $k \equiv 0, 15 \pmod{16}$.

If $k \equiv 0 \pmod{3}$, then $3|k$. If $k \equiv 1 \pmod{3}$, then $3|2k+1$. If $k \equiv 2 \pmod{3}$, then $3|k+1$.

Thus, there are no restrictions on $k$ in $\pmod{3}$.

It is easy to see that only one of $k$, $k+1$, and $2k+1$ is divisible by $5$. So either $k, k+1, 2k+1 \equiv 0 \pmod{25}$.

Thus, $k \equiv 0, 24, 12 \pmod{25}$.

From the Chinese Remainder Theorem, $k \equiv 0, 112, 224, 175, 287, 399 \pmod{400}$. Thus, the smallest positive integer $k$ is $\boxed{112}$.

Addition 9/23/17

To elaborate, we write out all 6 possibilities of parings. For example, we have

$k \equiv 24 \pmod{25}$ $k \equiv 15 \pmod{16}$

is one pairing, and

$k \equiv 24 \pmod{25}$ $k \equiv 0 \pmod{16}$

is another. We then solve this by writing the first as $16k+15 \equiv 24 \pmod{25}$ and then move the 15 to get $16k \equiv 9 \pmod{25}$.

We then list out all the mods of the multiples of 16, and realize that each of these 6 pairings can be generalized to become one of these multiples of 16.

The chain is as follows:

$16 \pmod{25}$ 7, 23, 14, 5, 21, 12, 3, 19, 10, 1, 17, 8, 24, 15, 6, 22, 13, 4, 20, 11, 27, 18, 9, 0, and then it loops.

We see that for the first equation we have 9 mod 25 at the 21st position, so we then do $24(16)+15$ to get the first answer of 399.

Again, this is possible to repeat for all 6 cases. CRT guarantees that we will have a solution before 25*16 or 400 and indeed we did :P

See also

2002 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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