maths
posted by kala .
a number which is less than or equal to 150 but
if counted in pairs, one will remain
if counted in threes, two will remain
if counted in fours, three will remain
if counted in fives, four will remain
if counted in sixes, five will remain
if counted in sevens, nothing will remain
what is the number?

This is a problem in number theory and related to the Chinese remainder theorem.
I do not know at what level you're working on. However, this problem is probably made to be solved without the use of advanced theory. Here's an approach that you can apply to other problems of the same nature.
We have the following conditions:
1. if counted in pairs, one will remain
2. if counted in threes, two will remain
3. if counted in fours, three will remain
4. if counted in fives, four will remain
5. if counted in sixes, five will remain
6. if counted in sevens, nothing will remain
Conditions 1 to 5 "happen" to be one less than a multiple of the counting number. Therefore, we can find a solution to conditions 1 to 5 by finding the LCM of the corresponding numbers 2,3,4,5,6 and subtract 1.
Since LCM(2,3,4,5,6)=3*4*5=60, we see immediately that 59 satisfies all of conditions 1 to 5. You should check this for yourself.
We can also see that for any positive integer k, k*LCM1 will also share the same property. For example, for k=3, 60*31=179 is also a solution to conditions 15. Also, check this for yourself.
We conclude therefore, to satisfy condition 6, we only need to find (by trial and error), a value of k such that 7 divides (60*k1), in which case (60k1) will be the required number.
I will leave it to you to complete the problem. Please post any time if you have questions. 
147
Respond to this Question
Similar Questions

bio
Determine the number of colonies counted on the LB plate in the following transformation experiment. The amount of plasmid on the plate being counted was 0.04 micrograms. The transformation efficiency was calculated to be 4 x 10^3. 
8th grade math
members of a nature club participate in an official butterfly count every summer. in summer of 1996 members counted 225 monarch butterflies . in summer of 1997 members counted 234 monarch butterflies. from 1996 to 1997 what was the … 
math
Richard put a counter on 6. He counted by tens. what are the next 3 numbers he counted? 
us history
read this quotation ¨of representation in congress is to be determined by the number of people who live in each state should slaves be counted? 
U.S. History
Read this question: "If representation in Congress is to be determined by the number of people who live in each state, should slaves be counted? 
U.S. History
Read this question: "If representation in Congress is to be determined by the number of people who live in each state, should slaves be counted? 
History
12.)"If representation in Congress is to be determined by the number of people who live in each state, should slaves be counted? 
Math
Carly and Kyle decided to count the number of red, blue, and yellow cars that they saw. They counted twice as many red cars as blue cars. The number of yellow cars they counted was one third of the number of red cars if they counted … 
Math Algrebra Word Problems People on a line
1) How many people are there between the 4,000th person from the front end and the 1,000th person from the back end when there are 4,000 people on a line? 
Math
Louisa has many marbles in a bag. When she counted them by twos, threes, fours, fives, sizes, sevens, eights, nines and tens there was always one left over. What was the smallest number of marbles Louisa could have had in her bag?