The number 13824

Q: Is 13824 a prime number?

No. 13824 = 2^9 × 3^3, so it is divisible by 2.

Q: What is special about the number 13824?

13824 is a perfect cube (24^3); a happy number: repeatedly adding the squares of its digits reaches 1; a Harshad number: divisible by its digit sum 18.

13824 is a perfect cube ( $24^{3}$ ). Its prime factorisation is $2^{9}$ × $3^{3}$ . Divisors (40): 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54 … 13824.

Number

Pick any whole number from 1 to 100,000.

13824 = 2^{9} \times 3^{3}

Explain it step by step in chat Make a short video of it

How to work it out

1
What is special about 13824: a perfect cube ( $24^{3}$ ); a happy number: repeatedly adding the squares of its digits reaches 1; a Harshad number: divisible by its digit sum 18.
2
13824 is composite. Its prime factorisation is
$13824 = 2^{9} \times 3^{3}$
3
The divisors add up to 40920; without 13824 itself that is 27096, so 13824 is abundant (27096 is more than 13824). Euler's totient φ(13824) = 4608 numbers up to 13824 share no factor with it.
4
The digits of 13824 add up to 18, and repeating that until one digit is left gives the digital root 9. The Collatz sequence (halve if even, else 3n+1) takes 120 steps to reach 1.

LaTeX source of the formulas on this page

\[ 13824 = 2^{9} \times 3^{3} \]

13824 at a glance

Property	Value
Type	composite
Number of divisors	40
Sum of divisors	40920
Euler totient φ(n)	4608
Square	191102976
Cube	2641807540224
Square root	117.575508 (= 48√6)
Digit sum / digital root	18 / 9
Collatz steps to 1	120
Scientific notation	1.3824 × $10^{4}$
Perfect power	$24^{3}$

13824 in other forms

Form	Value
In words	thirteen thousand eight hundred and twenty-four
Binary (base 2)	11011000000000
Octal (base 8)	33000
Hexadecimal (base 16)	3600
Base 3	200222000
As time	13824 seconds = 3 h 50 min 24 s

Is 13824 divisible by 2 to 12?

Divisor	Answer	Rule
2	Yes (13824 ÷ 2 = 6912)	A number is divisible by 2 if the last digit is even (0, 2, 4, 6, 8).
3	Yes (13824 ÷ 3 = 4608)	A number is divisible by 3 if the digits add up to a multiple of 3.
4	Yes (13824 ÷ 4 = 3456)	A number is divisible by 4 if the last two digits make a multiple of 4.
5	No (remainder 4)	A number is divisible by 5 if the last digit is 0 or 5.
6	Yes (13824 ÷ 6 = 2304)	A number is divisible by 6 if it is divisible by both 2 and 3.
7	No (remainder 6)	A number is divisible by 7 if double the last digit and subtract it from the rest: the result is a multiple of 7.
8	Yes (13824 ÷ 8 = 1728)	A number is divisible by 8 if the last three digits make a multiple of 8.
9	Yes (13824 ÷ 9 = 1536)	A number is divisible by 9 if the digits add up to a multiple of 9.
10	No (remainder 4)	A number is divisible by 10 if the last digit is 0.
11	No (remainder 8)	A number is divisible by 11 if the alternating sum of the digits is a multiple of 11.
12	Yes (13824 ÷ 12 = 1152)	A number is divisible by 12 if it is divisible by both 3 and 4.

Questions

Is 13824 a prime number?

No. 13824 = $2^{9}$ × $3^{3}$ , so it is divisible by 2.

What are the factors of 13824?

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 192, 216, 256, 288, 384, 432, 512, 576, 768, 864, 1152, 1536, 1728, 2304, 3456, 4608, 6912, 13824.

What is 13824 in binary?

11011000000000 (hexadecimal 3600, octal 33000).

What is the square root of 13824?

About 117.5755, exactly 48√6.

Is 13824 divisible by 3?

Yes. Its digits add up to 18, a multiple of 3.

What is special about the number 13824?

13824 is a perfect cube ( $24^{3}$ ); a happy number: repeatedly adding the squares of its digits reaches 1; a Harshad number: divisible by its digit sum 18.