Sixth Grade Math

A · Sixth Grade Math

In sixth grade, students are presented with different ways to calculate the Least Common Multiple
( LCM ) and the Greatest Common Factor ( GCF ) of two integers. The LCM of two integers a and b is
the smallest positive integer that is a multiple of both a and b. The GCF of two non-zero integers a
and b is the largest positive integer that divides both a and b without remainder.

For this problem you will write a program that determines both the LCM and GCF for positive
integers.

Input

The first line of input contains a single integer N, (1 ≤ N ≤ 1000) which is the number of data sets that
follow. Each data set consists of a single line of input containing two positive integers, a and b,
(1 ≤ a, b ≤ 1000) separated by a space.

Output

For each data set, you should generate one line of output with the following values: The data set
number as a decimal integer (start counting at one), a space, the LCM, a space, and the GCF.

Sample Input	Sample Output

B · Cryptoquote

A cryptoquote is a simple encoded message where one letter is simply replaced by another
throughout the message. For example:

Encoded: HPC PJVYMIY
Decoded: ACM CONTEST

In the example above, H=A, P=C, C=M, J=O, V=N, Y=T, M=E and I=S. For this problem, you will
decode messages.

Input

The first line of input contains a single integer N, (1 ≤ N ≤ 1000) which is the number of data sets that
follow. Each data set consists of two lines of input. The first line is the encoded message. The
second line is a 26 character string of upper case letters giving the character mapping for each letter
of the alphabet: the first character gives the mapping for A, the second for B and so on. Only upper
case letters will be used. Spaces may appear in the encoded message, and should be preserved in
the output string.

Output

For each data set, you should generate one line of output with the following values: The data set
number as a decimal integer (start counting at one), a space and the decoded message.

C · Binary Clock

A binary clock is a clock which displays traditional sexagesimal time (military format) in a binary
format. The most common binary clock uses three columns or three rows of LEDs to represent zeros
and ones. Each column (or row) represents a time-unit value.

When three columns are used (vertically), the bottom row in each column represents 1 (or 2⁰), with
each row above representing higher powers of two, up to 2⁵ (or 32). To read each individual unit
(hours, minutes or seconds) in the time, the user adds the values that each illuminated LED
represents, and then reads the time from left to right. The first column represents the hour, the next
column represents the minute, and the last column represents the second.

When three rows are used (horizontally), the right column in each row represents 1 (or 2⁰), with each
column left representing higher powers of two, up to 2⁵ (or 32). To read each individual unit (hours,
minutes or seconds) in the time, the user adds the values that each illuminated LED represents, and
then reads the time from top to bottom. The top row represents the hour, the next row represents the
minute, and the bottom row represents the second.

For example:

0 = LED off Vertically	1 = LED on Horizontally

Time is: 10 : 37 : 49

For this problem you will read a time in sexagesimal time format, and output both the vertical and
horizontal binary clock values. The output will be formed by concatenating together the bits in each
column (or row) to form two 18 character strings of 1’s and 0’s as shown below.

10:37:49 would be written vertically as 011001100010100011 and horizontally as
001010100101110001.

Input

The first line of input contains a single integer N, (1 ≤ N ≤ 1000) which is the number of data sets that
follow. Each data set consists of a single line of input containing the time in sexagesimal format.

Output

For each data set, you should generate one line of output with the following values: The data set
number as a decimal integer (start counting at one), a space, the binary time in vertical format (18
binary digits ), a space and the binary time in horizontal format (18 binary digits).

Sample Input	Sample Output

D · Recursively Palindromic Partitions

A partition of a positive integer N is a sequence of integers which sum to N, usually written with plus
signs between the numbers of the partition. For example

15 = 1+2+3+4+5 = 1+2+1+7+1+2+1

A partition is palindromic if it reads the same forward and backward. The first partition in the example
is not palindromic while the second is. If a partition containing m integers is palindromic, its left half is
the first floor(m/2) integers and its right half is the last floor(m/2) integers (which must be the
reverse of the left half. (floor(x) is the greatest integer less than or equal to x.)

A partition is recursively palindromic if it is palindromic and its left half is recursively palindromic or
empty. Note that every integer has at least two recursively palindromic partitions one consisting of all
ones and a second consisting of the integer itself. The second example above is also recursively
palindromic.

For example, the recursively palindromic partitions of 7 are:

7, 1+5+1, 2+3+2, 1+1+3+1+1, 3+1+3, 1+1+1+1+1+1+1

Write a program which takes as input an integer N and outputs the number of recursively palindromic
partitions of N.

Input

The first line of input contains a single integer N, (1 ≤ N ≤ 1000) which is the number of data sets that
follow. Each data set consists of a single line of input containing a single positive integer for which
the number of recursively palindromic partitions is to be found.

Output

For each data set, you should generate one line of output with the following values: The data set
number as a decimal integer (start counting at one), a space and the number of recursively
palindromic partitions of the input value.

Sample Input	Sample Output