NUMBER THEORY MATH CIRCLES
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Muhammad Yusuf
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1. How many 7-digit palindromic
numbers are there? How many 8-digit palindromic numbers are there?
2. How many 8-digit non-palindromic numbers are there?
3. The last palindromic year was
2002. How many years from now (2010) will the next palindromic year occur?
4. Break the following composite
numbers down into their prime factors using a factor tree.
(a) 24
(b) 80
(c) 144
5. I am a composite number
between 1 and 10. I am not a perfect square, but I have three prime factors.
What number am I?
6. a) Paul has forgotten his
locker combination which is a 5-digit palindromic number. He knows that the
fourth digit is 7. How many combinations does he have to try in order to find
the right one?
b) Paul remembers that his
combination is less than 30000. How many combinations does he have to try now?
7. You can create palindromic
numbers using the following method:
- Start with the number 3462 and reverse its digits.
- Add the two numbers together to get 3462 + 2643 = 6105
- Repeat the process with the new number (6105) until you get a palindromic number.
- You will get 6105 + 5016 = 11121, then 11121 + 12111 = 23232, which is a palindromic number.
Choose a different number and use
this method to get another palindromic number. Note: Although this method works
with most numbers, it is not certain that it works for all numbers. 196, 887,
1675 and 7436 are some examples of numbers that have yet to produce palindromic
numbers using this method.
8. How many palindromic numbers
are between 123 and 456?
9. Are there more palindromic
numbers or perfect squares under 100? Are there more palindromic numbers or
prime numbers under 100?
10. We know that a number can be
both a perfect square and palindromic, and both prime and palindromic. Can a
number be both a perfect square and prime?
11. Do the following
calculations:
a) 1 x 1 =
b) 11 x 11 =
c) 111 x 111 =
d) 1111 x 1111 =
What do you notice? Do you think
the pattern will continue?
Continue the pattern up to 111
111 111 x 111 111 111. What do you think
will happen to the when you square 1 111 111 111?
12. I am a prime number greater
than 100 and less than 1000. Each of my digits is different and also prime. I
am the smallest prime number such that my hundreds digit is greater than my
ones digit. What number am I?
13. I am a two-digit prime
number. The sum of my digits is 8. If you add me to 300, I become a palindromic
prime number. What number am I?
14. I am a 5-digit palindromic
perfect square. Each of my digits is also a perfect square and my last two
digits form a palindromic number. My middle digit is the largest of the 5
digits. What number am I?
15. An odd palindromic number has
an odd number of digits. Can you tell if the sum of its digits will be odd or
even?
16. Jessica is trying to figure
out 262 without a calculator. She knows that it is a 3-digit
palindromic number where the sum of the digits is 19 and the middle digit is
one more than the other two. Can you help her out?
17. Beth would like to visit her
friend Nathan who is living on the main street of a small town. The main street
has 50 houses divided into two blocks numbered from 1 to 20 and 21 to 50. Since
Beth has forgotten the number, she asks a passer-by, who replies, "Just
try to guess it." Beth likes playing games and asks three questions:
(a) In which block is it?
(b) Is the number even?
(c) Is it a perfect square?
After Beth has received the
answers, she says, "I'm still not sure which house it is, but if you tell
me whether the digit 4 is in the number, I will know the answer!" Then she
runs to the building in which she thinks Nathan is in. A lady opens the door
and it turns out she is wrong. The lady starts laughing and tells Beth,
"Your advisor is the biggest liar in the whole town. He never speaks the
truth!" Beth thinks for a moment and says, "Thanks, now I know Nathan's
real address." What is Nathan's address?
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