







Divisibility of 3n^5+7n 
20161214 

From Parag:
Question from Parag, a student:
if n is a natural number,then 3n^5+7n is divisible by
a)2
b)3
c)5
d)7
i got the answer but still i need a valid alternate approach. Answered by Penny Nom. 





4821x14y is an 8digit number divisible by 72 
20140806 

From RAYA: if 4821x14y is an 8digit number divisible by 72. How many values can x and y take? Answered by Penny Nom. 





The square of any odd number, decreased by 1, is divisible by 8 
20121116 

From bailey: Prove that the square of any odd number, decreased by 1, is divisible by 8 Answered by Penny Nom. 





The difference of the two numbers 
20100215 

From Steve: The difference of the two numbers 'abcdef ' and ' fdebca ' is divisible by 271. prove
that b = d and c = e. Answered by Claude Tardif. 





How many combinations of 8614 are divisible by 7? 
20080122 

From Rebecca: How many combinations of 8614 are divisible by 7 equally (with no remainder)? Answered by 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. 





Divisibility 
20070518 

From Ashish: A number is divisible by 2^n if the last n digits of the number are divisible by 2^n.
Why? Answered by Penny Nom and Claude Tardif. 





Divisibility by each of the first ten counting numbers 
20051017 

From Simon: determine smallest positive integer that is divisible by each of the first ten counting numbers Answered by Penny Nom. 





A 3 digit number divisible by 7 
20040503 

From A student: We need to arrange 1,3 and 6 to form a 3 digit number that is divisible by 7. Answered by Penny Nom. 





Divisibility by 2 or 5 or both 
20031030 

From Abdu: How many positive integers less than 1,001 are divisible by either 2 or 5 or both? Answered by Penny Nom. 





Three consecutive positive intergers 
20030209 

From Yew: Prove that when we multiply any consecutive positive intergers, the result is always divisible by 6.
ex. (7)(8)(9) = 504 = 6 (84) Answered by Penny Nom. 





Divisibility of 5^{ 2002} 
20020825 

From Simon: I need to ask you a question if 5^{ 2002} and 3^{ 2002} are divisible by 26. Answered by Penny Nom. 





The number of hidden cubes 
20020205 

From Katie: This problem is about finding the number of cubes visible and hidden in a cube.
In a cube that is 3x3, 19 cubes can be seen. 8 are hidden. In a cube that is 4x4, 37 cubes can be seen. 27 are hidden. In a cube that is 5x5, 61 cubes can be seen. 64 are hidden. In a cube that is 6x6, 91 cubes can be seen. 125 are hidden. The question is: Explain how you could find the number of small cubes that are visible and hidden in a cube of any size. Answered by Paul Betts and Penny Nom. 





Divisibility by 9 
20001024 

From Kelera: If the sum of the digits of a number is divisible by 9, then the number itself it divisible by 9. Why is that? How do you explain this? Answered by Penny Nom. 





111...1222...2 
19990811 

From Brad Goorman: Let N = 111...1222...2, where there are 1999 digits of 1 followed by 1999 digits of 2. Express N as the product of four integers, each of them greater than 1. Answered by Penny Nom. 





Divisibility of 2n choose n. 
19960924 

From Kathy Doan: Can you prove that "2n choose n" is not divisible by 3, 5, and 7 for infinitely many n? Answered by Penny Nom. 

