- Still trying to draw a bicycle!
- Pack squares with side \(1/2, 1/3, 1/4,...\) into the square with side \(1\).
- Find products of some triples of consecutive numbers. Are they divisible by \(6\)? Is this always so?
- What last digit of a product depends on? Find last digits of some pairs, like \(23\times 56\) and \(53\times 36\). Are they always the same? What is the last digit of \(847281937\times 938274\times 9629052717\)?
- How the remainder of division of \(n\) by \(10\) is related to the last digit of \(n\)? What is \(n\) is negative?
- Find the last digit of \(17^{17}\).
- In general, what happens with the remainders \(\mod p\) for some \(p\), like \(5\) or \(9\), when numbers are multiplied?
- Sets: intersections (AND) and unions (OR).
Homework:
- If one finds the sum of digits of \(17^{17}\), then sum of its digits, and so on, till one digit remains, - what is it?
- The following sets are given:\[A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\},
D = \{0, 7, 100, 333\}.
\]
Describe the sets: - \(A \cup B\);
- \(A\cap B\);
- \((A \cap B)\cup D\);
- \(C \cap (D\cap B)\);
- \((A\cup B)\cap (C \cup D)\);
- \((A\cap B)\cup (C \cap D)\);
- \((D\cup A)\cap (C \cup B)\);
- \((A\cap (B\cap C))\cap D\);
- \((A\cup (B\cap C))\cap D\);
- \((C \cap A)\cup ((A\cup (C \cap D))\cap B)\)
Reminder: no class on Sept. 29.
