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Question Number 19403 by Tinkutara last updated on 10/Aug/17

$$\mathrm{Let}P\left(n\right)=(n+1)(n+3)(n+5)(n+7)(n+9).$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{largest}\mathrm{integer}\mathrm{that}\mathrm{is}\mathrm{a}$$$$\mathrm{divisor}\mathrm{of}P\left(n\right)\mathrm{for}\mathrm{all}\mathrm{positive}\mathrm{even}$$$$\mathrm{integers}n?$$

Commented by RasheedSindhi last updated on 12/Aug/17

$$15$$

Answered by RasheedSindhi last updated on 12/Aug/17

$$\mathrm{P}\left(\mathrm{n}\right)=(\mathrm{n}+1)(\mathrm{n}+3)(\mathrm{n}+5)(\mathrm{n}+7)(\mathrm{n}+9)$$$$\mathrm{n}\in {\mathbb{E}}^{+};\mathrm{Let}\mathrm{n}=2\mathrm{m},\mathrm{m}\in \mathbb{N}$$$$\mathrm{P}\left(2\mathrm{m}\right)=(2\mathrm{m}+1)(2\mathrm{m}+3)(2\mathrm{m}+5)(2\mathrm{m}+7)(2\mathrm{m}+9)$$$$\mathrm{m}=3\mathrm{k}\mid \mathrm{m}=3\mathrm{k}+1\mid \mathrm{m}=3\mathrm{k}+2$$$$\mathrm{C}-0:\mathrm{m}=3\mathrm{k}\Rightarrow$$$$\mathrm{P}\left(\mathrm{n}\right)=\mathrm{P}\left(2\mathrm{m}\right)=\mathrm{P}(2.3\mathrm{k})$$$$=(6\mathrm{k}+1)(6\mathrm{k}+3)(6\mathrm{k}+5)(6\mathrm{k}+7)(6\mathrm{k}+9)$$$$=3(6\mathrm{k}+1)(2\mathrm{k}+1)(6\mathrm{k}+5)(6\mathrm{k}+7)(6\mathrm{k}+9)$$$$\mathrm{C}-1:\mathrm{m}=3\mathrm{k}+1\Rightarrow$$$$\mathrm{P}\left(\mathrm{n}\right)=(6\mathrm{k}+3)(6\mathrm{k}+5)(6\mathrm{k}+7)(6\mathrm{k}+9)(6\mathrm{k}+11)$$$$=9(2\mathrm{k}+1)(6\mathrm{k}+5)(6\mathrm{k}+7)(2\mathrm{k}+3)(6\mathrm{k}+11)$$$$\mathrm{C}-2:\mathrm{m}=3\mathrm{k}+2\Rightarrow$$$$\mathrm{P}\left(\mathrm{n}\right)=(6\mathrm{k}+5)(6\mathrm{k}+7)(6\mathrm{k}+9)(6\mathrm{k}+11)(6\mathrm{k}+13)$$$$=3(6\mathrm{k}+5)(6\mathrm{k}+7)(2\mathrm{k}+3)(6\mathrm{k}+11)(6\mathrm{k}+13)$$$$\mathrm{In}\mathrm{all}\mathrm{the}\mathrm{three}\mathrm{cases}:$$$$3\mid \mathrm{P}\left(\mathrm{n}\right)$$$$\mathrm{Similarly}\mathrm{we}\mathrm{can}\mathrm{prove}\mathrm{that}$$$$5\mid \mathrm{P}\left(\mathrm{n}\right)$$$$\mathrm{Also}\mathrm{can}\mathrm{be}\mathrm{verified}\mathrm{from}\mathrm{P}\left(10\right)$$$$\mathrm{that}3\mathrm{and}5\mathrm{can}\mathrm{divide}\mathrm{P}\left(\mathrm{n}\right)\mathrm{once}\mathrm{only}.$$$$\mathrm{P}\left(10\right)=11.13.15.17.19$$$$=11.13.3.5.17.19$$$$\mathrm{From}\mathrm{the}\mathrm{following}\mathrm{it}\mathrm{is}\mathrm{proved}$$$$\mathrm{that}\mathrm{there}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{any}\mathrm{divisor}\mathrm{of}\mathrm{P}\left(\mathrm{n}\right)$$$$\mathrm{other}\mathrm{than}3\mathrm{\&}5$$$$\mathrm{P}\left(2\right)=3.5.7.9.11$$$$\mathrm{P}\left(10\right)=11.13.15.17.19$$$$\mathrm{P}\left(12\right)=13.15.17.19.21$$$$\mathrm{Hence}\mathrm{the}\mathrm{largest}\mathrm{divisor}\mathrm{is}$$$$3\times 5=15$$

Commented by Tinkutara last updated on 12/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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