1. ## challenge questions

1.
A number has three digits and is equal to 12 times the sum of its digits what the number is?

Solution:

Assume three digit number = xyz, then

100x + 10y + z = 12(x + y + Z)

100x + 10y + z = 12x + 12y + 12Z

Solve to get

88x = 2y + 11z

Take x=1, then y=0 and z= 8 in order to satisfy the equation.

Three digit number is 108, hence 108 = 12(1+0+8).

Is there any other way to solve this problem?

2.

The sum of first N integers is a three digit number with all of its digits equal, what is the value of N?

1,2,3,4,5,6,7……………. Is an arithmetic series

S= n/2[2a + (n-1) d] = n (n +1) / 2

n (n +1) / 2 = 111x

Since sum of first integers is a three digit number with all digits equal
Hence it xxx = 100x +10x +x = 111x.

By trial and taking x=6, and solving for n, n2 +n – 1332 = 0, I will get n=36, is there any other way of solving;
n (n +1) / 2 = mod 111, and getting value of n=36 from this equation?

2. Originally Posted by nazz
1.
A number has three digits and is equal to 12 times the sum of its digits what the number is?
Solution:
Assume three digit number = xyz, then
100x + 10y + z = 12(x + y + Z)
100x + 10y + z = 12x + 12y + 12Z
Solve to get
88x = 2y + 11z
Take x=1, then y=0 and z= 8 in order to satisfy the equation.
Three digit number is 108, hence 108 = 12(1+0+8).
Is there any other way to solve this problem?
NO. 2 equations, 3 unknowns.

But you can "manipulate" such that answer becomes apparent:
11z = 88x - 2y
z = 8x - (2/11)y
Only y=0 is possible...get it?

3. Originally Posted by nazz
The sum of first N integers is a three digit number with all of its digits equal, what is the value of N?
1,2,3,4,5,6,7……………. Is an arithmetic series
S= n/2[2a + (n-1) d] = n (n +1) / 2
n (n +1) / 2 = 111x
Since sum of first integers is a three digit number with all digits equal
Hence it xxx = 100x +10x +x = 111x.
By trial and taking x=6, and solving for n, n2 +n – 1332 = 0, I will get n=36, is there any other way of solving;
n (n +1) / 2 = mod 111, and getting value of n=36 from this equation?
Use quadratic; discriminant will be sqrt(1 + 888x);
only 6 works; don't know if that's any faster...