48

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48 is a composite number, the smallest to have ten divisors (the nontrivial divisors being 2, 3, 4, 6, 8, 12, 16, 24, see A018261).

Membership in core sequences

Even numbers	..., 42, 44, 46, 48, 50, 52, 54, ...	A005843
Composite numbers	..., 44, 45, 46, 48, 49, 50, 51, ...	A002808
Abundant numbers	..., 36, 40, 42, 48, 54, 56, 60, ...	A005101
Number of rooted trees with $n$ nodes	..., 4, 9, 20, 48, 115, 286, 719, ...	A000081

Sequences pertaining to 48

Divisors of 48	1, 2, 3, 4, 6, 8, 12, 16, 24, 48	A018261
Multiples of 48	0, 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, 528, 576, ...

Partitions of 48

There are 147273 partitions of 48. Of these, the of the [FINISH WRITING]

Roots and powers of 48

In the table below, irrational numbers are given truncated to eight decimal places.

TABLE GOES HERE

Logarithms and 48th powers

REMARKS

TABLE

Values for number theoretic functions with 48 as an argument

TABLE GOES HERE

Factorization of 48 in some quadratic integer rings

As was mentioned above, 48 is the product of 2 4 and 3. But it has different factorizations in some quadratic integer rings.

$\mathbb {Z} [i]$	$(1\pm i)^{4}3$
$\mathbb {Z} [{\sqrt {-2}}]$	$({\sqrt {-2}})^{8}(1\pm {\sqrt {-2}})$	$\mathbb {Z} [{\sqrt {2}}]$	$({\sqrt {2}})^{8}3$
$\mathbb {Z} [\omega ]$	$(-1)2^{4}(1+2\omega )^{2}$	$\mathbb {Z} [{\sqrt {3}}]$	$(1\pm {\sqrt {3}})^{4}({\sqrt {3}})^{2}$
$\mathbb {Z} [{\sqrt {-5}}]$	2 4 × 3	$\mathbb {Z} [\phi ]$	2 4 × 3
$\mathbb {Z} [{\sqrt {-6}}]$		$\mathbb {Z} [{\sqrt {6}}]$	$(2\pm {\sqrt {6}})^{4}(3\pm {\sqrt {6}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$	$\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{4}3$	$\mathbb {Z} [{\sqrt {7}}]$	$(-1)(3\pm {\sqrt {7}})^{4}(2\pm {\sqrt {7}})$
$\mathbb {Z} [{\sqrt {-10}}]$	2 4 × 3	$\mathbb {Z} [{\sqrt {10}}]$	2 4 × 3
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$	$2^{4}\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)$	$\mathbb {Z} [{\sqrt {11}}]$	$(3\pm {\sqrt {11}})^{3}3$
$\mathbb {Z} [{\sqrt {-13}}]$	2 4 × 3	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$	$(-1)2^{4}\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)$
$\mathbb {Z} [{\sqrt {-14}}]$		$\mathbb {Z} [{\sqrt {14}}]$	$(4\pm {\sqrt {14}})^{4}3$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$		$\mathbb {Z} [{\sqrt {15}}]$	2 4 × 3
$\mathbb {Z} [{\sqrt {-17}}]$		${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$	$\left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{4}3$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$		$\mathbb {Z} [{\sqrt {19}}]$	$(-1)(13\pm 3{\sqrt {19}})^{4}(4\pm {\sqrt {19}})$

Representation of 48 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	110000	1210	300	143	120	66	60	53	48	44	40	39	36	33	30	2E	2C	2A	28

Notice that 48 is a Harshad number in every base from binary to base 19 except base 14.

See also

Some integers

							$-1$
0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
					1729