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Home / Elementary Number Theory / Show that any integer of the form 6k + 5 is also of the form, but not conversely.

Show that any integer of the form 6k + 5 is also of the form, but not conversely.

Solution: Let a is any integer and a = 6k + 5. We have to show that a is also of the form 3j + 2.

Now, a = 6k + 5

= 3.2k + 3 + 2

= 3(2k + 1) + 2

Let, j = 2k + 1

∴ a = 3j + 2

Conversely, if, a = 8 = 3.2+2

And a = 8 = 6.1 + 2

Here 1 and 2 are unique.

So, 8 ≠ 6k + 5.

Therefore, any integer of the form 6k + 5 is also of the form, but not conversely.               (Proved)

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