91Ó°ÊÓ

Each of the numbers $$ 1=1,3=1+2,6=1+2+3,10=1+2+3+4, \ldots $$ represents the number of dots that can be arranged evenly in an equilateral triangle: $$ \therefore \cdots $$ This led the ancient Greeks to call a number triangular if it is the sum of consecutive integers, beginning with 1. Prove the following facts concerning triangular numbers: (a) A number is triangular if and only if it is of the form \(n(n+1) / 2\) for some \(n \geq 1\). (Pythagoras, circa 550 B.C.) (b) The integer \(n\) is a triangular number if and only if \(8 n+1\) is a perfect square. (Plutarch, circa 100 A.D.) (c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus, circa 100 A.D.) (d) If \(n\) is a triangular number, then so are \(9 n+1,25 n+3\), and \(49 n+6\). (Euler, 1775 )

A farmer purchased 100 head of livestock for a total cost of \(\$ 4000\). Prices were as follow: calves, \(\$ 120\) each: lambs, \(\$ 50\) each; piglets, \(\$ 25\) each. If the farmer obtained at least one animal of each type, how many of each did he buy?

For an arbitrary integer \(a\), verify the following: (a) \(2 \mid a(a+1)\), and \(3 \mid a(a+1)(a+2)\). (b) \(3 \mid a\left(2 a^{2}+7\right)\) (c) If \(a\) is odd, then \(32 \mid\left(a^{2}+3\right)\left(a^{2}+7\right)\).

When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then noticed to his surprise that he had twice the amount of the original check. Determine the smallest value for which the check could have been written. [Hint: If \(x\) denotes the number of dollars and \(y\) the number of cents in the check, then \(100 y+x-68=2(100 x+y) .]\)

Solve each of the puzzle-problems below: (a) Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels, and each child \(\frac{1}{2}\) bushel. How many men, women, and children are there? (b) Mahaviracarya, 850 . There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers. What is the number of fruits in each pile? [Hint: Consider the Diophantine equation \(63 x+7=23 y\).] (c) Yen Kung, 1372. We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. How many coins are there? [Hint: If \(N\) is the number of coins, then \(N=77 x+27=78 y\) for integers \(x\) and \(y .]\) (d) Christoff Rudolff, \(1526 .\) Find the number of men, women, and children in a company of 20 persons if together they pay 20 coins, each man paying 3 , each woman 2, and each child \(\frac{1}{3}\) (e) Euler, 1770 . Divide 100 into two summands such that one is divisible by 7 and the other by 11 .

For a nonzero integer \(a\), show that ged \((a, 0)=|a|, \operatorname{gcd}(a, a)=|a|\), and \(\operatorname{gcd}(a, 1)=1\).

If \(a\) and \(b\) are integers, not both of which are zero, verify that $$ \operatorname{gcd}(a, b)=\operatorname{gcd}(-a, b)=\operatorname{gcd}(a,-b)=\operatorname{gcd}(-a,-b) $$

Given an odd integer \(a\), establish that $$ a^{2}+(a+2)^{2}+(a+4)^{2}+1 $$ is divisible by 12 .

Prove: The product of any three consecutive integers is divisible by 6 ; the product of any four consecutive integers is divisible by \(24 ;\) the product of any five consecutive integers is divisible by 120 . [Hint: See Corollary 2 to Theorem 2.4.]