645

The number 645 is a positive integer with various interesting mathematical properties, cultural references, and practical applications. Below is a detailed exploration of its characteristics.

Mathematical Properties

Prime Factorization:

645 is a composite number. Its prime factorization is:

[

645 = 3 \times 5 \times 43

]

This means it is the product of three distinct primes.

Divisibility:

645 has 8 divisors: 1, 3, 5, 15, 43, 129, 215, and 645. The sum of its proper divisors (excluding itself) is (1 + 3 + 5 + 15 + 43 + 129 + 215 = 411), making it deficient since 411 < 645.

Base Representations:

• Binary: (645_{10} = 1010000101_2)
• Octal: (645_{10} = 1205_8)
• Hexadecimal: (645_{10} = 285_{16})
• Roman numerals: DCXLV

Algebraic Properties:

645 is an odd number. It is not a perfect square (since (\sqrt{645} \approx 25.39685)), not a perfect cube, and not a prime number. It is a Harshad number (Niven number) in base 10 because the sum of its digits (6 + 4 + 5 = 15) divides 645: (645 \div 15 = 43), so it is divisible by its digit sum.

Other Number Theory Facts:

• 645 is a sphenic number because it is the product of three distinct primes.
• It is also a lucky number? Not sure; checking the lucky numbers list: 645 does not appear in the standard lucky numbers sequence (it is not lucky).
• It is a congruent number? A congruent number is a positive integer that represents the area of a right triangle with rational sides. 645 is congruent; one such triangle has sides ((24, 215/8, 1081/24))? Not sure, but likely.

Geometric and Numeric Insights

Area and Perimeter:

A rectangle with area 645 could have dimensions (1, 645), (3, 215), (5, 129), (15, 43). The rectangle with dimensions 15 by 43 has a perimeter of (2 \times (15 + 43) = 116), which is close to a perfect number (118? no).

Trigonometry:

The cosine of 645 degrees: (\cos(645^\circ) = \cos(645 - 720 = -75^\circ) = \cos(75^\circ) = 0.258819). The sine is (\sin(645^\circ) = \sin(-75^\circ) = -\sin(75^\circ) = -0.965926).

In Science and Technology

Astronomy:

• Asteroid 645 Agrippina is a minor planet in the asteroid belt, discovered in 1907.
• The number 645 might appear in astronomical catalogues or as a code for certain celestial objects.

Computing:

645 in hexadecimal (0x285) is used in color codes? #645 is not a standard web color, but #645000 is a shade of brown.

Physics:

• Atomic number 645 does not exist; the highest known atomic number is much lower.
• 645 nm is a wavelength of red-orange light, near the deep red end of the visible spectrum.

In Culture and History

Year 645:

In the Gregorian calendar, the year 645 was a common year starting on Wednesday. Notable events:

• In Japan, the Taika Reform began, centralizing government under the emperor.
• In the Byzantine Empire, conflicts with the Arabs continued.
• In China, the Tang dynasty was flourishing.

Other Uses:

• 645 is sometimes used as a model number for products (e.g., cameras, like the Pentax 645 medium format camera).
• In some area codes, 645 is not assigned; area code 645 is not in use in North America.
• In sports, jersey number 645 might be worn by a player, though uncommon.

Fun Facts and Miscellaneous

Puzzles:

• 645 can be expressed as the sum of consecutive integers: (645 = 214 + 215 + 216) (three consecutive numbers). Also, (645 = 106 + 107 + 108 + 109 + 110 + 105)? Not exactly. More systematically: It is the sum of 15 consecutive numbers: (35 + 36 + \dots + 49 = 645).
• It is the sum of two squares? (645 = 4^2 + 25^2 = 16 + 625 = 641), not 645. (645 = 2^2 + 25^2? 4+625=629), no. (645 = 14^2 + 23^2 = 196 + 529 = 725), too high. So it is not a sum of two squares. But it can be written as the sum of three squares: (645 = 1^2 + 10^2 + 24^2 = 1 + 100 + 576 = 677), no. Actually, (645 = 2^2 + 14^2 + 23^2 = 4+196+529=729), no. Wait: (645 = 8^2 + 10^2 + 23^2 = 64+100+529=693), no. I need to check: (645 = 11^2 + 14^2 + 20^2 = 121+196+400=717), no. (645 = 7^2 + 16^2 + 20^2 = 49+256+400=705), no. (645 = 10^2 + 17^2 + 18^2 = 100+289+324=713), no. So maybe it is not expressible as sum of three squares? Actually, every number not of the form (4^a(8b+7)) is a sum of three squares. 645 mod 8 = 645/8=80 remainder 5, so 880=640, remainder 5, so 645 = 880+5, not of the form 8b+7. And 645 is not divisible by 4 (since last two digits 45 not divisible by 4), so it should be expressible as three squares. Let's find: 645 - 23^2 = 645-529=116, which is 10^2+4^2=100+16, so 645 = 4^2+10^2+23^2. Yes! 4,10,23 works: 16+100+529=645. So it is a sum of three squares.
• 645 is a palindrome in base 4? 645 in base 4: 645/4=161 r1, 161/4=40 r1, 40/4=10 r0, 10/4=2 r2, 2/4=0 r2, so 22011_4, not palindrome.
• 645 in base 9: 645/9=71 r6, 71/9=7 r8, 7/9=0 r7, so 786_9, not palindrome.

Numerology:

In numerology, 645 reduces to (6+4+5=15), then (1+5=6). So its digital root is 6, which is associated with harmony, family, and responsibility.

Practical Applications

Everyday Life:

• 645 grams is a mass equivalent to about 1.42 pounds.
• 645 millimeters is 0.645 meters, or about 25.4 inches.
• 645 seconds is 10 minutes and 45 seconds.

Finance:

$645 could be a price, a salary, or a budget item. In ancient times, 645 pieces of silver might have been a fortune.

Conclusion

The number 645, while seemingly ordinary, reveals a rich tapestry of mathematical features and historical connections. From being a sphenic number to its appearance in the year 645 and its representation in various bases, it serves as an example of how numbers bridge abstract mathematics and tangible reality. Whether you encounter it in a math problem, a date, or a product model, 645 is more than just a sequence of digits—it's a gateway to exploration.