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n^2 is a multiple of 100 2015-03-30
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 2-digit number and a 3-digit number 2015-02-06
From Nathaniel:
The product of a 2-digit number and a 3-digit number is about 50 000 what are the products
Answered by Penny Nom.
Restricted partitions 2013-03-25
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.
The sum of digits of 4444^4444 2009-08-31
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 2009-08-06
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 2009-06-17
From Sophia:
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.

Answered by Robert Dawson.
GCD (a + b, a - b). 2009-04-01
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.
Some number theoretic speculations 2008-12-04
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^(p-1) - 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!)


Answered by Chris Fisher and Victoria West.
Finding the last non-zero digits of large factorials 2007-10-04
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 2007-08-24
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 2007-08-04
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 2007-05-07
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 2007-03-05
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 2007-02-07
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 2007-01-29
From Ankit:
11^n +22^n = 55^n

find the value of n?

Answered by Penny Nom.
1X2X3X4+1=5^5 2006-11-23
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 ... 2006-11-20
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 2006-05-05
From DeHayward:
Find the 6-digit 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 2006-04-08
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 2005-11-30
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+n-1 has no divisors ending with 3 or 7 2005-09-08
From Arne:
at least it seems like for any integers n and k,
10k+3 and 10k+7 do not divide nē+n-1
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. 2005-05-30
From Antonio:
The numbers p and 8p2 +1 are prime. Prove that the number 8p2+2p+1 is also a prime number.
Answered by Claude Tardif and Penny Nom.
Divisibility of a^2 + b^2 2005-05-16
From Ampa:
given natural numbers a and b such that a2+b2 is divisible by 21, prove that the same sum of squares is also divisible by 441.
Answered by Penny Nom.
39 consecutive natural numbers 2003-08-19
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 2002-11-20
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 2002-10-04
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? 2002-08-19
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 2002-04-09
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? 2001-01-10
From John:
Find all positive integers n so that n2 - 2 is a multiple of n - 4.
Answered by Sukanta Pati.
Divisibility by 16 2000-12-12
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 2000-12-10
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 2000-10-31
From A student:
I was wondering if you could help me answer a question my pre-algebra 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 2000-10-26
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? 2000-09-07
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.
n3 + 2n2 is a square 2000-09-04
From David Xiao:
determine the smallest positive integers, n , which satisfies the equation

n3 + 2n2 = b

where b is the square of an odd integer

Answered by Harley Weston.
Three consecutive odd integers 2000-08-18
From Wallace:
A six-digit 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 2000-07-05
From Harman Chaudhry:
Which is the smallest 10-digit number to be five times the cube of one number and also three times the fifth power of another?
Answered by Penny Nom.
Problems 2000-06-06
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.
  1. 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.
  2. 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? 2000-04-09
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.
The sum of the cubes is the square of the sum 1999-08-25
From Bernard Yuen:
How to prove 13 + 23 + 33 + 43 + ... n3 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 1998-11-27
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 1998-09-19
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)/(mn-1) an integer? 1997-02-11
From Ronald Lui:
Determine all ordered pairs (m,n) of positive integers such that ( n^3+1)/(mn-1)is an integer.
Answered by Richard McIntosh.



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