Mathematics
Senior Mathematical Challenge
2023
12

12

The greatest power of 7 which is a factor of 50! is 7^k (n! = 1×2×3×4×. . .× (n − 1) × n). What is k?

A. 4 B. 5 C. 6 D. 7 E. 8

Answer

E. 8

SOLUTIOn

50! = 1 × 2 × · · · × 50. Because the seven numbers 7, 14, 21, 28, 35, 42 and 49 are divisible by 7, they each contribute 1 to the power of 7 which is a factor 50! In addition, because 49 is divisible by 7 2 , it contributes an additional power of 7. Therefore the highest power of 7 that divides 50! is 7 + 1 = 8.

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