Overview
980 is a positive integer that lies between 979 and 981. As a natural number it is even, composite, and appears in everyday counting and numbering systems. In Roman numerals 980 is written CMLXXX.
Mathematical properties
The prime factorization of 980 is 2^2 × 5 × 7^2. From this factorization several basic arithmetic invariants follow: it has 18 positive divisors, Euler's totient function φ(980) = 336, and the sum of all positive divisors σ(980) = 2394. Because the sum of proper divisors (2394 − 980 = 1414) exceeds 980, the number is classified as an abundant number.
Divisors and factorizations
Breaking 980 into component factors highlights useful decompositions: 980 = 20 × 49 = 4 × 245 = 98 × 10. The complete list of positive divisors is:
- 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
Representations in other bases
980 can be expressed in common positional bases as follows. In binary it is 1111010100, in octal it appears as 1724, and in hexadecimal it is 3D4. Its decimal digit sum is 9 + 8 + 0 = 17, so it is not divisible by 9 and not a Harshad number in base 10.
Context, uses and distinctions
As a three-digit integer, 980 shows up in lists, identifiers and model numbers; it can also be a round figure in measurements or counts (for example, as a multiple of 20 or 49). Mathematically it is notable for combining a square prime factor (7^2) with a small even factor (2^2), yielding many divisors relative to its size. Historically, the string "980" can denote the year 980 CE, which falls in the early medieval period, but historical events and significance depend on the geographical and cultural context.
Notable arithmetic remarks
Because 980 = 4 × 5 × 49, it is divisible by 2, 4, 5, 10, 20, 49 and other listed divisors. It is not a perfect square or a perfect power (other than 7^2 dividing it), and it is not a triangular or perfect number. Its abundance (proper divisors sum 1414) measures how much the proper divisors exceed the number itself: 1414 − 980 = 434.