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16-10 Step-by-Step Examples
Show that the whole numbers k such that:
have digits in decimal notation.
We have:
so have digits in decimal notation.
Moreover, is divisible by 9, since its
decimal notation can only end in 9.
We also have:
and
so and are both divisible by 3.
Let’s consider whether B(3) is a prime number.
Type ISPRIME?(B(3))
and press . The
result is 1, which means
true. In other words, B(3) is
a prime.
Note: ISPRIME? is not
available from a CAS soft menu, but you can select it from
from CAS FUNCTIONS menu while you are in the
Equation Writer by pressing , choosing the
INTEGER menu, and scrolling to the ISPRIME? function.
To prove that is a prime number, it is
necessary to show that 1999 is not divisible by any of the
prime numbers less than or equal to . As
, that means testing the divisibility of
1999 by n = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41. 1999 is not divisible by any of these numbers, so we
can conclude that 1999 is prime.
10
n
k 10
n 1+
<≤ n 1+()
10
n
310
n
⋅ a
n
410
n
⋅ 10
n 1+
<<<<
10
n
b
n
210
n
⋅ 10
n 1+
<< <
10
n
210
n
⋅ c
n
310
n
⋅ 10
n 1+
<<<<
a
n
b
n
c
n
,, n 1+()
d
n
10
n
1–=
a
n
310
n
⋅ d
n
+=
c
n
310
n
⋅ d
n
–=
a
n
c
n
b
3
1999=
1999
1999 2025< 45
2
=
hp40g+.book Page 10 Friday, December 9, 2005 1:03 AM