9765625

9765625 is a ten‑digit integer equal to 5 to the 10th power. It can be expressed as 5^10 = (5^5)^2 = 3125^2, making it simultaneously a perfect fifth power, a tenth power, a square, and a power of 5. The number is derived by repeated multiplication of 5, and it equals 5, 25, 125, 625, 3125, and 15625 multiplied by 625 once more to reach 9,765,625. In base ten its digits are 9, 7, 6, 5, 6, 2, 5, which has no repeating pattern of obvious significance in decimal representation.

The prime factorization of 9765625 consists solely of the prime 5, occurring ten times; thus it is a prime power. The Euler totient function φ(9765625) equals 9765625 × (1 – 1/5) = 7,812,500. Because of its prime power form, many arithmetic functions related to divisors and totients have straightforward values. The divisor function τ(9765625) equals 11, as the number has 10 + 1 divisors. The sum‑of‑divisors function σ(9765625) equals (5^11 – 1) / (5 – 1) = 1,953,125.

In integer sequences, 9765625 occupies the 10th position of the sequence of powers of 5 (A000312) and the 5th position of square powers of 5 (A003158). It also assists in combinatorial constructions; for example, it counts the number of particular paths in a grid when combined with binomial coefficients involving powers of 5. Its representation in binary is 1001011000110010111, and in base 2 it has 19 bits. The number appears in computational contexts such as cryptographic hash functions and pseudorandom number generators that use powers of small primes for modulus values. The sheer size of 9765625, when compared with smaller powers of 5, illustrates the rapid growth of exponential sequences and their application in number theory.

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