Toán Lớp 7: Chứng minh rằng a)$16^{17}$ +$4^{33}$ +$8^{23}$ +$2^{67}$ chia hết cho 40 b)$81^{7}$ -$9^{13}$ +$12^{25}$ +$27^{9}$ -$12^{24}$ chia hế

Question

Toán Lớp 7:  Chứng minh rằng a)$16^{17}$ +$4^{33}$ +$8^{23}$ +$2^{67}$  chia hết cho 40 b)$81^{7}$ -$9^{13}$ +$12^{25}$ +$27^{9}$ -$12^{24}$ chia hết cho 11

tuanh · Answer

Lời giải và giải thích chi tiết:   a.Ta c&oacute;:   $A=16^{17}+4^{33}+8^{23}+2^{67}$   $ to A=(2^4)^{17}+(2^2)^{33}+(2^3)^{23}+2^{67}$   $ to A=2^{4 cdot 17}+2^{2 cdot 33}+2^{3 cdot 23}+2^{67}$   $ to A=2^{68}+2^{66}+2^{69}+2^{67}$   $ to A=2^{66}(2^2+1+2^3+2)$   $ to A=2^{66} cdot 15$   $ to A=2^{63+3} cdot 5 cdot 3$   $ to A=2^{63} cdot 2^3 cdot 5 cdot 3$   $ to A=2^{63} cdot 8 cdot 5 cdot 3$   $ to A=2^{63} cdot 40 cdot 3 quad vdots quad 40$   $ to đpcm$   b.Ta c&oacute;:   $B=81^7-9^{13}+12^{25}+27^9-12^{24}$   $ to B=(3^4)^7-(3^2)^{13}+12^{24} cdot 12+(3^3)^9-12^{24}$   $ to B=3^{28}-3^{26}+12^{24} cdot 12+3^{27}-12^{24}$   $ to B=3^{28}-3^{26}+3^{27}+12^{24} cdot 12-12^{24}$   $ to B=3^{26}(3^2-1+3)+12^{24} cdot (12-1)$   $ to B=3^{26} cdot 11+12^{24} cdot 11 quad vdots quad 11$   $ to đpcm$

maianhtuanh · Answer

Giải đáp:   a, 16^17+4^33+8^23+2^67   = 2^68 + 2^66 + 2^69 + 2^67    = 2^66.(2^3+2^2+2+1)   = 2^66 . 15   = 2^3 . 2^63 . 15   = 8.15.2^63   = 120 . 2^63   = 40.3.2^63  vdots 40   b, 81^7 - 9^13 + 12^25 + 27^9 - 12^24   = (3^4)^7 - (3^2)^13 + (2^2 .3)^25 + (3^3)^9 - (2^2 .3)^24   = 3^28 - 3^26 + 2^50 . 3^25 + 3^27 - 2^48 . 3^24   = 3^26.(3^2 + 3 - 1) + 2^48 . 3^24.( 2^2 . 3 - 1)   = 11 . (3^26 + 2^48 . 3^24)  vdots 11