AMC 10B 2019 q. 14

S

Sean Hannity

New member
Joined
Jan 29, 2019
Messages
3
No idea what they mean by base-ten representation or why the numbers are lined up in that certain order. Could someone explain this question me + how to do it?
Thanks :)

The base-ten representation for
$19!$
is
$121,6T5,100,40M,832,H00$
, where
$T$
,
$M$
, and
$H$
denote digits that are not given. What is
$T+M+H$
?
$\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17$
 
Sean Hannity said:
No idea what they mean by base-ten representation or why the numbers are lined up in that certain order. Could someone explain this question me + how to do it?
Thanks :)

The base-ten representation for
$19!$
is
$121,6T5,100,40M,832,H00$
, where
$T$
,
$M$
, and
$H$
denote digits that are not given. What is
$T+M+H$
?
$\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17$
Click to expand...
Do you understand what does 19! represent?
 
Sean Hannity said:
No idea what they mean by base-ten representation or why the numbers are lined up in that certain order. Could someone explain this question me + how to do it?
Thanks :)

The base-ten representation for
$19!$
is
$121,6T5,100,40M,832,H00$
, where
$T$
,
$M$
, and
$H$
denote digits that are not given. What is
$T+M+H$
?
$\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17$
Click to expand...
Base-ten representation means writing out a number as we usually do, such as 12,345 meaning twelve-thousand, three hundred forty-five. Each digit represents the number of some power of ten: 1 ten-thousand, 2 thousands, 3 hundreds, 4 tens, and 5 ones.

The symbol 19!, read "19 factorial", means [MATH]19\times18\times...\times2\times1[/MATH].

They are saying it can be written out as 121,6_5,100,40_,832,_00, where the three blanks are represented by the letters T, M, and H.

The question asks you for the sum of those three digits.

To find the answer, you have to use whatever you know about divisibility.

If you have no idea what to do now, then perhaps this problem is above your current knowledge. Are you expected to solve it?
 
The base 10 representation of numbers is what you learned in first grade: it uses place value and the TEN digits called Arabic numerals, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The other representation of numbers often taught in grade school are Roman numerals, but they have no practical use today.. In modern practical work, binary representation, with digits 0 and 1, and hexadecimal representation, with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F, are sometimes used and are place-value systems like the decimal system (base 10) system. So in hexadecimal, 12 means 1 * 16 + 2 * 1 = 18 in decimal.

They are just telling you here that you are NOT dealing with some representation other than the one you are familiar with.
 
Dr.Peterson said:
Base-ten representation means writing out a number as we usually do, such as 12,345 meaning twelve-thousand, three hundred forty-five. Each digit represents the number of some power of ten: 1 ten-thousand, 2 thousands, 3 hundreds, 4 tens, and 5 ones.

The symbol 19!, read "19 factorial", means [MATH]19\times18\times...\times2\times1[/MATH].

They are saying it can be written out as 121,6_5,100,40_,832,_00, where the three blanks are represented by the letters T, M, and H.

The question asks you for the sum of those three digits.

To find the answer, you have to use whatever you know about divisibility.

If you have no idea what to do now, then perhaps this problem is above your current knowledge. Are you expected to solve it?
Click to expand...
No not really lol, we don't do these types of problems at school. I'm just trying to get better at math so I can at least do somewhat well in math contests. I looked at the answer and I'm confused why they chose specifically the numbers 11 and 9. ---> "We can figure out H = 0 by noticing that 19! will end with 3 zeroes, as there are three 5s in its prime factorization. Next, we use the fact that 19! is a multiple of both 11 and 9."
 
Please show us the answer you looked at, so we can discuss it accurately. (I found the problem in several places, one of which is probably what you are referring to.)

But essentially, this is a matter of knowing your divisibility tests, and realizing that those for 9 and 11 are convenient for this purpose. Math contests commonly expect both breadth of knowledge and some level of creativity.

Do you know those tests?
 
Dr.Peterson said:
Please show us the answer you looked at, so we can discuss it accurately. (I found the problem in several places, one of which is probably what you are referring to.)

But essentially, this is a matter of knowing your divisibility tests, and realizing that those for 9 and 11 are convenient for this purpose. Math contests commonly expect both breadth of knowledge and some level of creativity.

Do you know those tests?
Click to expand...

Art of Problem Solving

artofproblemsolving.com artofproblemsolving.com
 
That's one of the sites I found.

The first solution shown uses modular arithmetic notation; that can be very useful, so you should learn it if you don't.

The second solution is more explicit about the divisibility tests they used. Are you familiar with those (involving sums of digits, changing signs in the case of 11)?

Do you see why 19! must be a multiple of 9 and of 11? Do you see why these two tests are easier to apply here than most others?
 
You must log in or register to reply here.
Top