Download 2009-2010 Meet 1

SOML MEET 1
EVENT 3
Number Sense
1.
[2 Points]
NAME: __________________
TEAM: __________________
SCHOOL: __________________
How many numbers less than 60 are prime? Remember: “1” is not prime.
ANS:
2.
[3 Points]
_
If a < b , then 32 + 4 2 + 52 + 12 2 = a 2 + b 2 is satisfied by only one pair of
positive integers ( a, b ) . What is the value of a + b ?
ANS: _______
3.
[5 Points]
What would be the units (ones) digit for the number 39999 ?
Hint: Look for a pattern in the ones digit of the powers of 3.
ANS: ________________
SOML MEET 1
EVENT 3
Number Sense
1.
[2 Points]
KEY
How many numbers less than 60 are prime? Remember: “1” is not prime.
Solution: Primes less than 60: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
ANS: 17
2.
[3 Points]
Solution:
If a < b , then 32 + 42 + 52 + 122 = a 2 + b 2 is satisfied by only one pair of
positive integers ( a, b ) . What is the value of a + b ?
(3
2
) (
)
+ ( 5 + 12 ) =a
+ 42 + 52 + 122 =a 2 + b 2
52
2
2
2
+ b2
52 + 169 =a 2 + b 2
52 + 132 =a 2 + b 2
a = 5 b=13
a+b = 5+13 = 18
ANS: 18
3.
[5 Points]
What would be the units (ones) digit for the number 39999 ?
Hint: Look for a pattern in the ones digit of the powers of 3.
Solution: Look for a pattern in the in the ones digit of the powers of 3:
=
31 3=
35 243=
39 19683
=
32 9=
36 729 =
310 59049
=
33 27
=
37 2187=
311 177147
=
34 81
=
38 6561=
312 531441
If the exponent is equal to 1 or has a remainder of 1 when divided by 4, then the ones digit is a 3.
If the exponent is equal to 2 or has a remainder of 2 when divided by 4, then the ones digit is a 9.
If the exponent is equal to 3 or has a remainder of 3 when divided by 4, then the ones digit is a 7.
If the exponent has a zero remainder when divided by 4, then the ones digit is a 1.
For the number 39999 , we divide the exponent 9999 by 4 to get 2499.75 or 2499 3 . Thus the
4
exponent has a remainder of 3 upon division by 4, so the ones digit for the number 39999 is 7.
ANS:
7