# 112

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## Number

$112$ (one hundred and twelve) is:

$2^4 \times 7$

The $7$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$:
$112 = 1 + 7 + 11 + 16 + 21 + 26 + 31 = \dfrac {7 \left({5 \times 7 - 3}\right)} 2$

The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $8$ different ways

The $16$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$:
$112 = 56 \times 2 = 56 \times \left({1 \times 1 \times 2}\right)$

The $30$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $112$, $\ldots$

The $53$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$ which cannot be expressed as the sum of distinct pentagonal numbers

The length of the side of the smallest perfect square dissection of an integer square

The side length of the smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex

## Historical Note

$112$ is the number of pounds avoirdupois in a hundredweight.