State which of the statement in True or False. The expression 79 … - Vidyadip

Question

State which of the statement in True or False.
The expression $79 + 97$ is divisible by $64.$
Hint: $79 + 97 = (1 + 8)^7 - (1 – 8)^9$

✓

Answer

True.Solution:
$7^9+9^7=(1+8)^7-(1-8)^9$
$=[\ ^7\text{C}_0+\ ^7\text{C}_1.8+\ ^7\text{C}_2(8)_2+\ ^7\text{C}_3(8)^3+...+\ ^7\text{C}_7(8)^7]\\-[\ ^9\text{C}_0+\ ^9\text{C}_18+\ ^9\text{C}_28+\ ^9\text{C}_3(8 )^2-\ ^9\text{C}_3(8)^3+...\ ^9\text{C}_9(8)^9]$
$=(7\times8+9\times8)+(21\times8^2-36\times8^2)+...$
$=(56+72)+(21-36)8^2+...=128+64(21-36)+...$
$=64[2+(21-36)+...]$
Which is divisible by $64$
Hence, the given statement is True.

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