# 1120 There’s Lots of Shade under These Factor Trees!

It’s hot this summer, but you can rest under the shade of some of 1120’s factor trees. Whether you want a tall tree or a wide one, these are only a few of the MANY possible ones you can choose. Every one of them gives you the same prime factors which are shown in pink here. Here are some things I’ve learned about the number 1120:

• 1120 is a composite number.
• Prime factorization: 1120 = 2 × 2 × 2 × 2 × 2 × 5 × 7, which can be written 1120 = 2⁵ × 5 × 7
• The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 1120 has exactly 28 factors.
• Factors of 1120: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, 1120
• Factor pairs: 1120 = 1 × 1120, 2 × 560, 4 × 280, 5 × 224, 7 × 160, 8 × 140, 10 × 112, 14 × 80, 16 × 70, 20 × 56, 28 × 40, or 32 × 35
• Taking the factor pair with the largest square number factor, we get √1120 = (√16)(√70) = 4√70 ≈ 33.4664 1120 is the sum of the twenty prime numbers from 17 to 101.
It is also the sum of these two consecutive primes:
557 + 563 = 1120

1120 is the hypotenuse of exactly one Pythagorean triple:
672-896-1120 which is (3-4-5) times 224

I like the way 1120 looks in these other bases:
It’s 1112111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 2(3³) + 3² + 3  + 1 = 1120,
929 in BASE 11 because 9(11²) + 2(11) + 9(1) = 1120,
WW in BASE 34 (W is 32 base 10) because 32(34) + 32(1) = 32(35) = 1120,
and W0 in BASE 35 because 32(35) = 1120

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