It is true that the expression [tex](36^5-6^9)(38^9-38^8)[/tex] is divisible by 30 and 37
How to prove the expression
The expression is given as:
[tex](36^5-6^9)(38^9-38^8)[/tex]
Express 36 as 6^2
[tex]((6^2)^5-6^9)(38^9-38^8)[/tex]
This gives
[tex](6^{10}-6^9)(38^9-38^8)[/tex]
Factor out 6^8
[tex]6^8(6^2-6)(38^9-38^8)[/tex]
Evaluate the exponent of 2
[tex]6^8(36-6)(38^9-38^8)[/tex]
Evaluate the difference
[tex]6^8(30)(38^9-38^8)[/tex]
Factor out 38^8
[tex]6^8(30)(38^8)(38-1)[/tex]
[tex]6^8(30)(38^8)(37)[/tex]
30 and 37 are among the factors of the above expression.
Hence, it is true that the expression [tex](36^5-6^9)(38^9-38^8)[/tex] is divisible by 30 and 37
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