Difference between revisions of "2006 AMC 10B Problems/Problem 25"

Latest revision as of 13:39, 3 December 2023

Problem

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?

$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8$

Solutions

Solution 1

Let $S$ be the set of the ages of Mr. Jones' children (in other words $i \in S$ if Mr. Jones has a child who is $i$ years old). Then $|S| = 8$ and $9 \in S$ . Let $m$ be the positive integer seen on the license plate. Since at least one of $4$ or $8$ is contained in $S$ , we have $4 | m$ .

We would like to prove that $5 \not\in S$ , so for the sake of contradiction, let us suppose that $5 \in S$ . Then $5\cdot 4 = 20 | m$ so the units digit of $m$ is $0$ . Since the number has two distinct digits, each appearing twice, another digit of $m$ must be $0$ . Since Mr. Jones can't be $00$ years old, the last two digits can't be $00$ . Therefore $m$ must be of the form $d0d0$ , where $d$ is a digit. Since $m$ is divisible by $9$ , the sum of the digits of $m$ must be divisible by $9$ (see Divisibility rules for 9). Hence $9 | 2d$ which implies $d = 9$ . But $m = 9090$ is not divisible by $4$ , contradiction. So $5 \not\in S$ and $5$ is not the age of one of Mr. Jones' kids. $\boxed{ B }$

(We might like to check that there does, indeed, exist such a positive integer $m$ . If $5$ is not an age of one of Mr. Jones' kids, then the license plate's number must be a multiple of $lcm(1,2,3,4,6,7,8,9) = 504$ . Since $11\cdot504 = 5544$ and $5544$ is the only $4$ digit multiple of $504$ that fits all the conditions of the license plate's number, the license plate's number is $5544$ .)

Solution 2

Alternatively, we can see that if one of Mr. Jones' children is of the age 5, then the license plate will have to end in the digit $0$ . The license plate cannot end in the digit $5$ as $2$ is a factor of the number, so it must be even. This means that the license plate would have to have two $0$ digits, and would either be of the form $XX00$ or $X0X0$ (X being the other digit in the license plate) . The condition $XX00$ is impossible as Mr. Jones can't be $00$ years old. If we separate the second condition, $X0X0$ into its prime factors, we get $X\cdot 10\cdot 101$ . $101$ is prime, and therefore can't account for allowing $X0X0$ being evenly divisible by the childrens' ages. The $10$ accounts for the 5 and one factor of 2. This leaves $X$ , but no single digit number contains all the prime factors besides 5 and 2 of the childrens' ages, thus $5$ $\boxed{ B }$ can't be one of the childrens' ages.

Solution 3

Another way to do the problem is by the process of elimination. The only possible correct choices are the highest powers of each prime, $2^3=8$ , $3^2=9$ , $5^1=5$ , and $7^1=7$ , since indivisibility by any powers lower than these means indivisibility by a higher power of the prime (for example, indivisibility by $2^2=4$ means indivisibility by $2^3=8)$ . Since the number is divisible by $9$ , it is not the answer, and the digits have to add up to $9$ . Let $(a,b)$ be the digits: since $2\cdot(a+b) | 9$ , $(a+b)|9$ . The only possible choices for $(a,b)$ are $(0,9)$ , $(1,8)$ , $(2,7)$ , $(3,6)$ , and $(4,5)$ . The only number that works is $5544$ , but it is not divisible by $5$ . $\boxed{ B }$

~edited by mobius247

Solution 4

Since 5 has the "strictest" divisibility rules of the single digit numbers, we check that case first. To be divisible by 9, the other repeating digit must be 4. Seeing that 5544 is the only arrangement of 2 4's and 2 5's that is divisible by the other single digit numbers and itself is not divisible by 5, the answer is $\boxed {(B) 5}$ . -liu4505 ~edited by mobius247

Video Solution

https://www.youtube.com/watch?v=BigiBRo3G2c ~David

@@ Line 4: / Line 4: @@
 <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math>
-== Solution ==
+== Solutions ==
-Since all the children have different ages, and the oldest child is <math>9</math>, only <math>1</math> of the numbers between <math>1</math> and <math>8</math> inclusive cannot be the age of a child.
+=== Solution 1===
+Let <math>S</math> be the set of the ages of Mr. Jones' children (in other words <math>i \in S</math> if Mr. Jones has a child who is <math>i</math> years old). Then <math>|S| = 8</math> and <math>9 \in S</math>. Let <math>m</math> be the positive integer seen on the license plate. Since at least one of <math>4</math> or <math>8</math> is contained in <math>S</math>, we have <math>4 | m</math>.
-To be evenly divisible by the age of all the kids, the license plate number must be a multiple of the [[least common multiple]] of the ages of all the kids.
+We would like to prove that <math>5 \not\in S</math>, so for the sake of contradiction, let us suppose that <math>5 \in S</math>. Then <math>5\cdot 4 = 20 | m</math> so the units digit of <math>m</math> is <math>0</math>. Since the number has two distinct digits, each appearing twice, another digit of <math>m</math> must be <math>0</math>. Since Mr. Jones can't be <math>00</math> years old, the last two digits can't be <math>00</math>. Therefore <math>m</math> must be of the form <math>d0d0</math>, where <math>d</math> is a digit. Since <math>m</math> is divisible by <math>9</math>, the sum of the digits of <math>m</math> must be divisible by <math>9</math> (see [[Divisibility_rules/Rule_for_3_and_9_proof | Divisibility rules for 9]]). Hence <math>9 | 2d</math> which implies <math>d = 9</math>. But <math>m = 9090</math> is not divisible by <math>4</math>, contradiction. So <math>5 \not\in S</math> and <math>5</math> is not the age of one of Mr. Jones' kids. <math>\boxed{ B }</math>
-Since being divisible by <math>5</math> restricts the possibilities of the digits in the number, let's analyze the possibilities of license plate numbers if one of Mr. Jones's kids is <math>5</math> years old.
+(We might like to check that there does, indeed, exist such a positive integer <math>m</math>. If <math>5</math> is not an age of one of Mr. Jones' kids, then the license plate's number must be a multiple of <math> lcm(1,2,3,4,6,7,8,9) = 504 </math>. Since <math>11\cdot504 = 5544</math> and <math>5544</math> is the only <math>4</math> digit multiple of <math>504</math> that fits all the conditions of the license plate's number, the license plate's number is <math>5544</math>.)
-Since the age of one of Mr. Jones's kids must be an even number, the number must be divisible by <math>10</math>.
+=== Solution 2 ===
+Alternatively, we can see that if one of Mr. Jones' children is of the age 5, then the license plate will have to end in the digit <math>0</math> . The license plate cannot end in the digit <math>5</math> as <math>2</math> is a factor of the number, so it must be even. This means that the license plate would have to have two <math>0</math> digits, and would either be of the form <math>XX00</math> or <math>X0X0</math> (X being the other digit in the license plate) . The condition <math>XX00</math> is impossible as Mr. Jones can't be <math>00</math> years old. If we separate the second condition, <math>X0X0</math> into its prime factors, we get <math> X\cdot 10\cdot 101 </math>. <math>101</math> is prime, and therefore can't account for allowing <math>X0X0</math> being evenly divisible by the childrens' ages.  The <math>10</math> accounts for the 5 and one factor of 2. This leaves <math>X</math>, but no single digit number contains all the prime factors besides 5 and 2 of the childrens' ages, thus <math>5</math> <math>\boxed{ B }</math> can't be one of the childrens' ages.
-Since the number is divisible by <math>10</math>, the units digit must be <math>0</math>
+=== Solution 3 ===
+Another way to do the problem is by the process of elimination. The only possible correct choices are the highest powers of each prime, <math>2^3=8</math>, <math>3^2=9</math>, <math>5^1=5</math>, and <math>7^1=7</math>, since indivisibility by any powers lower than these means indivisibility by a higher power of the prime (for example, indivisibility by <math>2^2=4</math> means indivisibility by <math>2^3=8)</math>. Since the number is divisible by <math>9</math>, it is not the answer, and the digits have to add up to <math>9</math>. Let <math>(a,b)</math> be the digits: since <math>2\cdot(a+b) | 9</math>, <math>(a+b)|9</math>. The only possible choices for <math>(a,b)</math> are <math>(0,9)</math>, <math>(1,8)</math>, <math>(2,7)</math>, <math>(3,6)</math>, and <math>(4,5)</math>. The only number that works is <math>5544</math>, but it is not divisible by <math>5</math>. <math>\boxed{ B }</math>
-Since the number has two distinct digits, each appearing twice, another digit must be <math>0</math>
+~edited by mobius247
-Since Mr. Jones can't be <math>00</math> years old, the last two digits can't be <math>00</math>
+=== Solution 4 ===
-Therefore, the number must be in the form <math>d0d0</math>, where <math>d</math> is a digit.
+Since 5 has the "strictest" divisibility rules of the single digit numbers, we check that case first. To be divisible by 9, the other repeating digit must be 4. Seeing that 5544 is the only arrangement of 2 4's and 2 5's that is divisible by the other single digit numbers and itself is not divisible by 5, the answer is <math>\boxed {(B) 5}</math>.
+-liu4505
+~edited by mobius247
-Since the oldest child is <math>9</math>, the number must be divisible by <math>9</math>.
+== Video Solution ==
-By [[divisibility rules]], the sum of the digits must be a multiple of <math>9</math>
+https://www.youtube.com/watch?v=BigiBRo3G2c    ~David
-<math> 2d \equiv 0\bmod{9} </math>
+== See also ==
+{{AMC10 box|year=2006|ab=B|num-b=24|after=This is the last question}}
+{{AMC12 box|year=2006|ab=B|num-b=18|num-a=20}}
-<math> d = 0 </math> or <math> d = 9 </math>
+[[Category:Introductory Number Theory Problems]]
+{{MAA Notice}}
-Since the number has two distinct digits, <math>d</math> can't be <math>0</math>.
-So <math> d = 9 </math>, and the only possible number is <math>9090</math>.
-Since either <math>4</math> or <math>8</math> is one of the kids' ages, the number must be divisible by <math>4</math>.
-Since <math>9090</math> is not divisible by <math>4</math>, it is not a possible number.
-Therefore, there are no possible license plate numbers if <math>5</math> is an age of one of Mr. Jones's kids. So <math>5</math> can't be the age of one of Mr. Jones's kids.
-If <math>5</math> is not an age of one of Mr. Jones's kids, then the license plate number must be a multiple of <math> lcm(1,2,3,4,6,7,8,9) = 504 </math>.
-Since <math>11\cdot504 = 5544</math> and <math>5544</math> is the only <math>4</math> digit multiple of <math>504</math> that fits all the conditions of the license plate's number, the license plate's number is <math>5544</math> and the number that is not an age of one of Mr. Jones's kids is <math> 5 \Rightarrow B </math>
-== See Also ==
-*[[2006 AMC 10B Problems]]
-*[[2006 AMC 10B Problems/Problem 24|Previous Problem]]

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