HP 40gs User Manual page 284

hp40g+.book Page 10 Friday, December 9, 2005 1:03 AM
16-10
Show that the whole numbers k such that:
n n + 1
≤ < (
10 k 10 n
have
We have:
n n
< ⋅ < <
10 3 10 a 4 10
n
n n
< < ⋅
10 b 2 10
n
n n
< ⋅ < <
10 2 10 c 3 10
n
( )
a , b , c n + 1
so have
n n n
n
d = 10 – 1
Moreover,
n
decimal notation can only end in 9.
We also have:
n
⋅
a = 3 10 + d
n n
and
n
⋅
c = 3 10 – d
n n
a c
so and are both divisible by 3.
n n
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
INTEGER menu, and scrolling to the ISPRIME? function.
b = 1999
To prove that
3
necessary to show that 1999 is not divisible by any of the
prime numbers less than or equal to
2
<
1999 2025 = 45
, 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.
)
+ 1
digits in decimal notation.
n n + 1
⋅ <
10
n + 1
<
10
n n + 1
⋅ <
10
digits in decimal notation.
is divisible by 9, since its
, choosing the
is a prime number, it is
1999
. As
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