[log3 , where the brackets designate “the largest integer in the number” (see the note just prior to problem 101). 5. One block is placed on each pan (first weight trial). There are two possible outcomes: On the first weight trial one of the pans is heavier. In this event one of the blocks must be aluminum and the other duraluminum. We now place both blocks on one pan and weigh them against pairs of remaining blocks (those being divided into nine pairs arbitrarily). Any pair of blocks which

relatively prime to 125; for such an x0, x1 is an integer such that x0 + 125 is divisible by x1 (that is, x1 is a divisor of , and is divisible by x0. Since , we may use the fact that x1 is the smaller of the neighbors of x0 to obtain . It is not difficult to verify that all pairs x0, x1 satisfying the conditions of the problem can be expressed as follows: . This yields the following solution arrays: . All possible pairs of relatively prime positive integers x, y, less than 1000, such that

set x = 1 and use the identity we obtain from which it follows that It is proved in the same manner that (b) We can obtain the required result by reference to either the first or the second solution of problem (a). We shall not repeat those solutions here, but we shall derive the formulas we need from the formulas of problem (a). Since it follows that [see problem (a)]. If we divide this formula by and use the identities We obtain Similarly, we have But Therefore, for odd n (n = 2k +

and the other of the form – y2 – 1. But if then that is, x2 + y2 = mp is divisible by p. Remark: The problem could have required that neither of the integers x or y is to exceed p/2, that is, that the sum x2 + y2 + 1 be less than p2 and the quotient m resulting from the division of x2 + y2 + 1 by p be less than p. 249. (a) The assertion of the problem follows from the following identity: the validity of which can readily be verified. Remark: Since the identity just displayed is rather

sum of the divisors of the integer n. 104. Does there exist a natural number n such that the fractional part of the number , that is, the difference , exceeds 0.999999? 105.* (a) Prove that for any natural number n, the integer is odd. (b) Find the highest power of 2 which divides the integer . 106. Prove that if p is an odd prime, it divides the difference . 107.* Prove that if p is a prime number, the difference is divisible by p. ( is the number of combinations of n