$41\cdot 42\cdot 43\cdot \ldots \cdot 144\cdot 145$ ko'paytma 5 ning qanday eng katta darajasiga bo'linadi?
- A. 62
- B. 24
- C. 16
- D. 26
Bo'linuvchanlik. Sonning natural bo'luvchilar soni va yig'indisi bo‘yicha savol
Javob va yechimni ko‘rish
Javob: 26
Ko‘paytma $41\cdot42\cdot\ldots\cdot145$ dagi har bir sonning $5$ ga bo‘linish darajasini (ya’ni $5$ ning qancha kuchi bo‘linishini) hisoblaymiz.
Bu daraja $v_{5}(n!)$ ga teng bo‘ladi, chunki $41\cdot42\cdot\ldots\cdot145=\dfrac{145!}{40!}$.
Legendre formulasidan
\[
v_{5}(145!)=\Big\lfloor\frac{145}{5}\Big\rfloor+\Big\lfloor\frac{145}{25}\Big\rfloor+\Big\lfloor\frac{145}{125}\Big\rfloor=29+5+1=35,
\]
\[
v_{5}(40!)=\Big\lfloor\frac{40}{5}\Big\rfloor+\Big\lfloor\frac{40}{25}\Big\rfloor=8+1=9.
\]
Shu sababli
\[
v_{5}\!\left(\frac{145!}{40!}\right)=v_{5}(145!)-v_{5}(40!)=35-9=26.
\]
Demak, $41\cdot42\cdot\ldots\cdot145$ ko‘paytma $5^{26}$ ga bo‘linadi, lekin $5^{27}$ ga bo‘linmaydi. Eng katta daraja $26$, ya’ni javob **D**.
Mavzuni mustahkamlang
Bu savol Bo'linuvchanlik. Sonning natural bo'luvchilar soni va yig'indisi mavzusiga tegishli. Ta’riflar, formulalar va misollarni mavzu sahifasida ko‘ring.
Mavzuni o‘rganish