







A question about the empty set 
20180617 

From Andrey: Hello there!
I got that an empty set is a subset of every set.
There is a question.
Is an empty set an element of every set?
∅ ⊆ {x}True
∅ ∈{x}?
Sorry if the question is easy. A set theory is a bit confusing. Answered by Penny Nom. 





n^2 is a multiple of 100 
20150330 

From Rahul: I have to prove that n^2 is a multiple of 100 is necessary or
Sufficient condition (or both) for n being multiple of 10 Answered by Penny Nom. 





The product of a 2digit number and a 3digit number 
20150206 

From Nathaniel: The product of a 2digit number and a 3digit number is about 50 000 what are the products Answered by Penny Nom. 





A union and interception problem 
20131019 

From Zakir: Sir, I have some problem in union there is a Question in a book
find B set if A={2,4,6,8} , AUB={2,4,5,6,7,8} and A intersection B={6,8}
plz tell me how can I find the B set Answered by Penny Nom. 





Restricted partitions 
20130325 

From vidya: I am having a series of numbers eg.( 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
I can take any 5 digits eg(15,10,8,6,5) and it should not repeat and the summation should be any predefined static value . eg(44)
That is (15+10+8+6+5=44) . How many summation series will result 44 ?
My problem is how to find this using a formula or any other simpler automation method is there instead of checking one by one all the combinations.
Plz do help me... Thnks in advance Answered by Chris Fisher. 





Prove A intersect B =X iff A = X and B = X 
20100306 

From Gloria: how would you prove A intersect B =X iff A = X and B = X Answered by Tyler Wood. 





The sum of digits of 4444^4444 
20090831 

From SHIVDEEP: The sum of digits of 4444^4444 is A .The sum of digits of A is B .
Find the sum of digits of B ? Answered by Claude Tardif. 





30 students 
20090806 

From Peter: Question from Peter, a student:
Can you please help with the following questions?
(a) It is known that among any group of the three students in a class two of them are friends. Total number of students is 25. Prove that there is a student who has at least 12 friends.
(b) There are 30 students in a class. They sit at 15 double desks, each one is for two students. Half of the girls sit with boys. Is it possible to make a rearrangement so that half of the boys sit with girls? Answered by Penny Nom. 





Divisibility 
20090617 

From Sophia: Hello
Please help my son with the solutions to the following:
a) Determine the remainder when 2^2009 + 1 is divided by 17;
b) Prove that 30^99 + 61^100 is divisible by 31;
c) It is known that numbers p and 8p^2+1 are primes. Find p.
Again, your assistance is greatly appreciated.
Thanks
Sophia Answered by Robert Dawson. 





GCD (a + b, a  b). 
20090401 

From Tomas: Let a and b integer and relatively prime. Prove that:
GCD (a + b , a  b) = 1 or 2 Answered by Stephen La Rocque. 





More on infinity and Set Theory 
20090217 

From Justin: I greatly appreciate your help I was just wondering from your previous answer, why doesn't Cantor's cardinal numbers in set theory apply to the limit x>0, y=infinity?
Justin Answered by Robert Dawson. 





Infinity and Set Theory 
20090217 

From Justin: I was just wondering is the limit x>0, y=1/x=infinity, the biggest uncountable infinity according to Cantor's cardinal numbers in set theory?
Justin Answered by Robert Dawson. 





Some number theoretic speculations 
20081204 

From Andrew: Another way of looking at the 'alternating parity polynomial', again based on Fermat's Little Theorem,
is to substitute (a  b) for x in x^(p1)  1 as this is always divisible by any prime p. So, if one removes
the " 1", there is always a remainder of (1/p)! (I took up your challenge!)
. . .
Andrew Answered by Chris Fisher and Victoria West. 





Finding the last nonzero digits of large factorials 
20071004 

From Mukesh: i have to find last five non zero digits of integer which can be very large (
upto 10^12) . i can find last non zero digit of of any factorial. Now my problem is that
i have to find last five non zero digit of factorial and also i want to general method for
last K non zero digits of factorial n. For example 10!=3628800 so last non zero digit is 8 ,last two
non zero digit is 88 .....and last five non zero digit is 36288. Answered by Victoria West. 





A question about integers 
20070824 

From Jerry: Does there exist a positive integer such that when it is written in base 10 and its leftmost digit is crossed out, the new number is 56 times less than the original number? Answered by Stephen La Rocque and Penny Nom. 





Induction  divisibility 
20070804 

From Jerry: How would you prove that for any positive integer n, the value of the expression 3^(2n+2)  8n 9 is divisible by 64. Answered by Chris Fisher and Penny Nom. 





A problem involving squarefree integers 
20070507 

From Andrew: I was told that if x > y (integers); then x would never exactly =
divide y^n (n integer > 1) if (x,y) =3D 1 ; or if x is "square =
free". Is the latter true and why? Answered by Stephen La Rocque and Penny Nom. 





The tens digit of a really large number 
20070305 

From Sai: How can i find the tens digit of a really large number? i was gven 63^15 + 15^63 in a competitive exam. Answered by Penny Nom. 





An even positive integer cubed minus four times the number 
20070207 

From Rachael: I can't figure out the proof or the method to get the proof for this question: any even positive integer cubed minus four times the number is divisible by 48 Answered by Haley Ess and Penny Nom. 





11^n +22^n = 55^n 
20070129 

From Ankit: 11^n +22^n = 55^n
find the value of n? Answered by Penny Nom. 





1X2X3X4+1=5^5 
20061123 

From Liza: 1X2X3X4+1=5 square 2x3x4x5+1=11 square What is the rule for this? Answered by Stephen La Rocque and Penny Nom. 





Pick any prime number greater than 3,square it ,then ... 
20061120 

From Eliseo: I was ask to pick any prime number greater than 3,square it ,then subtract 4, then divide the new result by 12 and record the remainder. He told me the remainder was 9. How could he be sure that the remainder was 9 without knowing which prime I picked? Answered by Stephen La Rocque. 





A number theory problem 
20060505 

From DeHayward: Find the 6digit number in which the first digit is one less than the second, the third is half of the second, the fourth is 3 times the third, and the last 2 digits are the sum of the fourth and fifth. The sum of all the digits is 24. Answered by Paul Betts. 





The sum of a three digit number and its three individual digits is 429 
20060408 

From Megan: Gill has recently moved to a new house, which has a three digit number. the sum of this number and its three individual digits is 429. What is the product of the three digits which make up the house number? Answered by Chris Fisher. 





Tables with perfect squares 
20051130 

From Craig: A table consists of eleven columns. Reading across the first row of the table we find the numbers 1991, 1992, 1993,..., 2000, 2001. In the other rows, each entry in the table is 13 greater than the entry above it, and the table continues indefinitely. If a vertical column is chosen at random, then the probability of that column containing a perfect square is: Answered by Claude Tardif. 





n^2+n1 has no divisors ending with 3 or 7 
20050908 

From Arne: at least it seems like for any integers n and k,
10k+3 and 10k+7 do not divide nē+n1
I tested this for every n from 0 to 3200 (which means same for the numbers from 3201 to 1)
could this be true, or is it just coincidence, or am I just totally wrong? Answered by Richard McIntosh. 





The numbers p and 8p^2 +1 are prime. 
20050530 

From Antonio: The numbers p and 8p^{2} +1 are prime. Prove that the number 8p^{2}+2p+1 is also a prime number. Answered by Claude Tardif and Penny Nom. 





Divisibility of a^2 + b^2 
20050516 

From Ampa: given natural numbers a and b such that a^{2}+b^{2} is divisible by 21, prove that the same sum of squares is also divisible by 441. Answered by Penny Nom. 





Matrices 
20031205 

From Julie: I am doing a project and need to find some mathematiciens who had an influence in matrices. I can't seem to find any when I search online. Could you please help me with this? Answered by Judi McDonald. 





39 consecutive natural numbers 
20030819 

From A student: Prove that among any 39 consecutive natural numbers it is always possible to find one whose sum of digits is divisible by 11. Answered by Penny Nom. 





abc,abc 
20021120 

From Pam: Prove or disprove that "every number of the form abc,abc (where a, b, and c represent digits) is divisible by 7,11, and 13" Answered by Penny Nom. 





Relatively prime 
20021004 

From Natasha: I really need help with this middle level math question. My little brother is asking me and I have no clue what the answer is. Explain what it means when two numbers are negatively prime. (?) Answered by Kathy Nolan and Penny Nom. 





When is 1! + 2! + 3! + ... + x! a square? 
20020819 

From Sarathy: Solve : 1! + 2! + 3! + ... + x! = y^{ 2} How do i find the solutions ? Answered by Claude tardif. 





n +1, n+2, n+3, and n+4 are all composite 
20020409 

From Jonathan: Find the small integer n such that n +1, n+2, n+3, and n+4 are all composite Answered by Penny Nom. 





Is n^2  2 a multiple of n  4? 
20010110 

From John: Find all positive integers n so that n^{2}  2 is a multiple of n  4. Answered by Sukanta Pati. 





Divisibility by 16 
20001212 

From Shiling: A number can be divided by 16 if and only if its 1st four digits can be divided by 16. How can you prove that? Answered by Penny Nom. 





A chemist had 8 flasks 
20001210 

From Jimmy: A chemist had 8 flasks capable of holding 12, 15, 27, 35, 37, 40, 53 and 69 fluid ounces respectively. He filled some with water and then filled all the rest except one with alcohol. He used exactly one and a half times as much alcohol as water. Which flask was left empty and which were left with water and which with alcohol? Answered by Claude Tardif. 





Perfect numbers 
20001031 

From A student: I was wondering if you could help me answer a question my prealgebra teacher asked in class the other day. He asked if we knew what the perfect numbers where. He told us the first number is 6 the second number is 28 but the third he did not tell us. Do you know what the third perfect number is? Answered by Paul Betts and Chris Fisher. 





A zip code problem 
20001026 

From Rob Mathis: Find the zip code of a place in a county so that the product of it and the zip code of another place in another county of the same name, but in a different state, is an exact multiple of the number 123456789 Answered by Claude Tardif. 





How many 17's and 19's total 1000? 
20000907 

From Jonathan: My question is: what 2 numbers would multiply 17 and 19 for a total of 1000. The numbers should not contain any decimal. Answered by Penny Nom. 





n^{3} + 2n^{2} is a square 
20000904 

From David Xiao: determine the smallest positive integers, n , which satisfies the equation n^{3} + 2n^{2} = b where b is the square of an odd integer Answered by Harley Weston. 





Three consecutive odd integers 
20000818 

From Wallace: A sixdigit integer XYXYXY, where X and Y are digits is equal to five times the product of three consecutive odd integers. Determine these three odd integers. Answered by Paul Betts. 





Five times a cube equals three times a fifth power 
20000705 

From Harman Chaudhry: Which is the smallest 10digit number to be five times the cube of one number and also three times the fifth power of another? Answered by Penny Nom. 





Problems 
20000606 

From Debbie Cummins: I am a Mum of a 12 yr.boy needing help with some math problems. I need not only the answers but how it is worked out.  Both the leftmost digit & the rightmost digit of a 4 digit number N are equal to 1. When these digits are removed, the 2 digit number thus obtained is N div by 21 Find N.
 Find all 3 digit even numbers N such that 693xN is a perfect square, that is, 693x N = k x k where k is an integer.
. . . Answered by Paul Betts and Claude Tardif. 





How many zeros? 
20000409 

From Greg Potts: The natural numbers 1 to 25 are multiplied together (1 x 2 x 3x..24 x 25). How many zeros are there in the product of this multiplication? a)6 b)7 c)5 d)10 or e)4? Answered by Harley Weston. 





Party favors 
20000222 

From Krystina Fernandez: Luanne was making party favors for her little sister's birthday party. Each party favor was in the shape of a cube. Luanne had pink and purple paint to paint the cubes and she could paint each face only solid pink or solid purple (no dots,stripes,ect.).For example, one cube may be all purple, another may have two purple faces and four pink faces. Her little sister wanted to have a different cube for each guest.(A cube is not considered different if it can be turned so that all it's sides match the corresponding sides of another cube.)How many different cubes was it possible for Luanne to make? Answered by Claude Tardif. 





Crossing number 
19991106 

From Christian: The crossing number of a graph G, denoted cr(G) is defined to be the minimum number of (pairwise) crossings of edges among all drawings of the graph in the plane. For example, cr(K5)=1 and cr(K3,3)=1. What is cr(K7,7)? I figured out that the answer is 81. Now I am trying to figure out if K7,7 can be drawn in the plane with less than 81 crossings? I'm not sure how to approach this one. Other than actually drawing it out and checking by trial and error, I am not sure how to approach this problem. Please help! Answered by Denis Hanson. 





The sum of the cubes is the square of the sum 
19990825 

From Bernard Yuen: How to prove 1^{3} + 2^{3} + 3^{3} + 4^{3} + ... n^{3} is equal to (1+2+3+...n)^{2}? (for n is positive integer) Answered by Harley Weston. 





Pick any odd number, square it, and then divide it by 8 
19981127 

From Brenda Meagher: Pick any odd number, square it, and then divide it by 8. No matter what odd number is chosen and squared and divided by 8, the remainder is one. Could you please explain this to me or is there a pattern that I am not aware Answered by Harley Weston. 





Five Factors 
19980919 

From Derek Yau: To whom it may concern, I have difficulty in getting the solution to the following question: Find 5 numbers that have exactly 5 factors. I got 16, 81 but couldn't find the rest. I believe that in order to have 5 factors, it has to be a square number. Isn't it true? I guess there may be a pattern to this. Thanks for your help. Derek Yau. Answered by Penny Nom. 





When is ( n^3+1)/(mn1) an integer? 
19970211 

From Ronald Lui: Determine all ordered pairs (m,n) of positive integers such that ( n^3+1)/(mn1)is an integer. Answered by Richard McIntosh. 

