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Ta có:
\(24^{54}.54^{24}.2^{10}=\left(2^3.3\right)^{54}.\left(3^3.2\right)^{24}.2^{10}\)
\(=\left(2^3\right)^{54}.\left(3^3.2\right)^{24}.2^{10}\)
\(=2^{162}.3^{54}.3^{72}.2^{24}.2^{10}\)
\(=2^{196}.3^{126}\) (1)
Lại có:
\(72^{63}=\left(2^3.3^2\right)^{63}=2^{189}.3^{126}\)(2)
Từ (1) và (2) ⇒ \(24^{54}.54^{24}.2^{10}⋮72^{63}\)
a, \(81^7-27^9-9^{13}\)
\(=3^{28}-3^{27}-3^{26}\)
\(=3^{22}\left(3^6-3^5-3^4\right)\)
\(=3^{22}\times405⋮405\)
\(2^{54}.54^{24}.2^{10}\)chia hết \(72^{63} \)
\(2^{54}.54^{24}.2^{10}\)=\((2^3.3)^{54}.(3^3.2)^{24}.2^{10}\)
=\((2^3)^{54}.3^{54}.(3^3)^{24}.2^{24}2^{10}\)
= \(2^{162}.2^{24}.2^{10}.3^{54}.3^{72}
\)
=\(2^{196}.3^{126}\)
\(72^{63}
\)=\((2^3.3^2)^{63}\)
=\((2^3)^{63}.(3^2)^{63}=2^{189}.3^{126}\)
Vì \(2^{196}.3^{126}\)chia hết \(2^{189}.3^{126}\)\(24^{54}.54^{24}.2^{10}\)
\(\Rightarrow \)\(24^{54}.54^{24}.2^{10}\)chia hết \(72^{63}
\)(dpcm)
\(24^{54}.54^{24}.2^{10}=3^{54}.2^{162}.2^{24}.3^{72}.2^{10}=3^{126}.2^{196}\)
ta có: \(72^{63}=9^{63}.8^{63}=\left(3^2\right)^{63}.\left(2^3\right)^{63}=3^{72}.2^{108}\)
ta có: \(\frac{3^{126}.2^{196}}{3^{72}.2^{108}}=3^{54}.2^{88}\)
suy ra \(3^{126}.2^{196}\) chia hết cho \(3^{72}.2^{108}\)
suy ra \(24^{54}.54^{24}.2^{10}\) chia hết cho \(72^{63}\)
\(\frac{1}{99}-\frac{1}{99.98}-\frac{1}{98.97}-....-\frac{1}{3.2}\)
=\(\frac{1}{99}-\left(\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{98.99}\right)\)
=\(\frac{1}{99}-\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{98}-\frac{1}{99}\right)\)
=\(\frac{1}{99}-\left(\frac{1}{2}-\frac{1}{99}\right)\)
=\(\frac{1}{99}-\frac{97}{198}\)
=\(\frac{-95}{198}\)
\(24^{54}.54^{24}.2^{10}\)
\(=\left(2^3.3\right)^{54}.\left(3^3.2\right)^{24}.2^{10}\)
\(=\left(2^3\right)^{54}.3^{54}.\left(3^3\right)^{24}.2^{24}.2^{10}\)
\(=2^{162}.3^{54}.3^{72}.2^{24}.2^{10}\)
\(=2^{196}.3^{126}\)
Lại có :
\(72^{63}=\left(2^3.3^2\right)^{63}\)
\(=\left(2^3\right)^{63}.\left(3^2\right)^{63}\)
\(=2^{189}.3^{126}\)
Vì \(2^{196}.3^{126}⋮2^{189}.3^{126}\)
\(\Leftrightarrowđpcm\)
77^6+7^5-7^4
=7^6.11^6+7^5-7^4
=7^4.7^2+7^4.7-7^4.1.11^6
=7^4.(7^2+7-1).11^6 chia hết cho 7
77^6+7^5-7^4 chia hết vì có số 7^4=7.7^3
Ta có 2454.5424.210=(23.3)54.(33.2)24.210=2162.354.372.224.210=2196.3126=(2189.3126).27=7263.27chia hết cho 7263(vì 7263chia hết cho 7263) => đpcm
\(24^{54}.54^{24}.2^{10}=\left(2^3\right)^{54}.3^{54}.2^{24}.\left(3^3\right)^{24}.2^{10}=2^{196}.3^{126}=2^7.2^{189}.\left(3^2\right)^{63}\)
\(=2^7.\left(2^3\right)^{63}.9^{63}=2^7.8^{63}.9^{63}=2^7.72^{63}\) chia hết cho \(72^{63}\)
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