Mnahattan Challenge Problem of the week! - 21/03/07

For a three-digit number xyz, where x, y, and z are the digits of the number, the function f(xyz) = 5^x2^y3^z. If f(abc) = 3 * f(def), what is the value of abc - def?

(A) 1
(B) 2
(C) 3
(D) 9
(E) 27


Answer : A
It is given in the question that .

It is obvious that the digits b, c, e, f are integers from 0 to 9 whereas that the digits a and d are integers from 1 to 9 (can't be equal to 0 because they are in the hundreds place).
For the above statement to be true, must be equal to , and .
Hence .

As the only difference between abc and def is in the units digits, the difference between these three-digit numbers is equal to , or 1.

Thus the right answer is A.

Manhattan Challenge Problem of the week! - 7 march 07

The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Answer - E

The product pq factors: {1, p, q, and pq}. If pq has no additional factors, then positive integers p and q must be prime. Pay heed to the fact that neither p nor q can be equal to 1; otherwise, pq would not have 4 distinct factors.

Statement (1) -- the sum of p and q is an odd integer. Therefore either p is even and q is odd, or p is odd and q is even. Since we know that both p and q are prime, either p or q must be equal to 2 (the only even prime number). But we do not know which of the two integers is equal to 2.

Statement (2) -- q is less than p -- gives us no information about the value of p

Taking both statements together, we can conclude that, because 2 is the smallest prime number, q must equal 2. But, we cannot determine the value of p.

Problem Solving - 6

For every positive integer n, the function h(n) is defined to be the product of all of the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

A. between 2 and 10

B. between 10 and 20

C. between 20 and 30

D. between 30 and 40

E. greater than 40

Answer - E

h(n)=2*4*6....100 =2(1*2*3*........50)

h(n) + 1 = 2(1*2*...50) + 1

Prime No ={2,3,........}

h(n) contains multiples of all the primes uptil 50 .... Hence E

OR

Let X = 2*4*6*.....*100

then X contains multiples of all the primes uptil 50
eg.
17 * 2 = 34 is in X.

37 * 2 = 74 is in X

Hence all the primes between 2 - 50 will be a factor of X => that none of them will be a factor of (X+1), hence the smallest prime should be at least greater than 50

Hence E.