Difference between revisions of "2003 AMC 10B Problems/Problem 25"

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Problem

What is the greatest integer $a,$ such that $a^2 + 3b$ is less than $(2b)^2,$ assuming that $b$ is $5?$

Solution

Solution 1

To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repeat the process until you can tell whether it is a multiple of three. This is the basis for the solution. Since the last two digits are $23$ , the sum of the digits is $2+3 = 5$ (therefore it is not divisible by three). However certain numbers can be added to make the sum of the digits a multiple of three.

$5+1 = 6$ ,

$5+4 = 9$ , and so on.

However since the largest four-digit number ending with $23$ is $9923$ , the maximum sum is

$5+18 = 23$ .

Using that process we can fairly quickly compile a list of the sum of the first two digits of the number.

$\{1, 4, 7, 10, 13, 16\}$

Now we find all the two-digit numbers that have any of the sums shown above. We can do this by listing all the two digit numbers $xy$ in separate cases.

$I. x+y = 1, \{10\} = 1$ $II. x+y = 4, \{13, 22, 31, 40\} = 4$ $III. x+y = 7, \{16, 25, 34, 43, 52, 61, 70\} = 7$ $IV. x+y = 10, \{19, 28, 37, 46, 55, 64, 73, 82, 91\} = 9$ $V. x+y = 13, \{49, 58, 67, 76, 85, 94\} = 6$ $VI. x+y = 16, \{79, 88, 97\} = 3$

And finally, we add the number of elements in each set.

$1+4+7+9+6+3 = \boxed{\textbf{(B)}\ 30}$

Solution 2

A number divisible by $3$ has all its digits add to a multiple of $3.$ The last two digits are $2$ and $3$ and add up to $5 \equiv 2\ (\text{mod}\ 3).$ Therefore the first two digits must add up to $1\ (\text{mod}\ 3).$ $4$ digits (including $0$ ) are $0\ (\text{mod}\ 3),$ $3$ are $1\ (\text{mod}\ 3),$ and $3$ are $2\ (\text{mod}\ 3).$ The following combinations are equivalent to $1\ (\text{mod}\ 3)$ :

$0\ (\text{mod}\ 3)+1\ (\text{mod}\ 3) \equiv 1\ (\text{mod}\ 3)+0\ (\text{mod}\ 3) \equiv 2\ (\text{mod}\ 3) +2\ (\text{mod}\ 3)$

Let the first term in each combination represent the thousands digit and the second term represent the hundreds digit. We can use this to find the total amount of four-digit numbers.

$3\cdot3 + 3\cdot4 + 3\cdot3 = 9 + 12 + 9 = \boxed{\textbf{(B)}\ 30}$

Solution 3

We have the following: $n \equiv 0 \pmod{3}$ and $n \equiv 23\pmod{100}$ . Then $n = 3a = 100b+23$ for some integers $a$ and $b$ . Taking mod 3 gives: $0 \equiv b+2 \pmod 3 \implies b\equiv 1\pmod 3$ so $b=3c+1$ for some integer $c$ . But $N = 100b+23 = 100(3c+1)+23$ so $n = 300c+123$ . Bounding this gives us: $999 < 300c+123 < 10000$ so $876 < 300c < 9877$ . Dividing by $300$ gives $2.92 < c < 32.9222$ so $3\le c \le 32$ . This gives $32-3+1 = \boxed{\textbf{(B)} \ 30 }$

Solution 4(extremely fast)

The number is in the form $xy23$

Notice that the number is divisible by $3$ if the sum of the digits of $xy23$ is divisible by $3$

Our first number that is divisible by $3$ is $1023$ , next is $1323$ .. Notice $xy$ goes from $10\implies 97$

Hence, there are $\frac{97-10}{3}+1=30$ distinct four digit numbers.

@@ Line 1: / Line 1: @@
 ==Problem==
-How many distinct four-digit numbers are divisible by <math> 3 </math> and have <math> 23 </math> as their last two digits?
+What is the greatest integer <math>a,</math> such that <math>a^2 + 3b</math> is less than <math>(2b)^2,</math> assuming that <math>b</math> is <math>5?</math>
-<math> \textbf{(A) }27\qquad\textbf{(B) }30\qquad\textbf{(C) }33\qquad\textbf{(D) }81\qquad\textbf{(E) }90 </math>
 ==Solution==

2003 AMC 10B (Problems • Answer Key • Resources)
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