What is the Sixth Digit of a Number That is a Multiple of 73 and 137?

[eightsquare]

Homework Statement

An eight digit number is a multiple of 73 and 137. If the second digit from the left of the number is seven, find the 6th digit from the left of the number.

Homework Equations

N.A.

The Attempt at a Solution

I don't know any clear method for solving this problem. I raised 73 to the 4th power and got an eight digit number, but it didn't have 7 as the second digit. Since 73 is very small compared to an eight digit number, it is impractical to write down all the eight digit numbers which are multiples of 73 and then finding one which is common with 137. Moreover, this problem is from an Olympiad and needs to be solved within roughly 72 seconds. I thought about LCM next but once I got the LCM(10001), I had no idea what to do with it. The HCF is 1(both numbers being prime). I substituted the numbers for variables. We get a7cdefgh. We need to find f. The problem is that I just don't see a way to solve this problem.

 

[sjb-2812]

What is 10001 x 1, x 2, x 3, .. x 12, x 13 etc. Do you see a pattern between the second digit and the sixth digits from the left of the number?

 

[eightsquare]

Oh i see it now. Thank you. So is a multiple of the lcm of two numbers divisible by each of the two numbers as a rule?

 

[Ray Vickson]

Oh i see it now. Thank you. So is a multiple of the lcm of two numbers divisible by each of the two numbers as a rule?

What do you think is meant by"lcm"?

 

[micromass]

Oh i see it now. Thank you. So is a multiple of the lcm of two numbers divisible by each of the two numbers as a rule?

Yes, the least common multiple of two numbers is (by definition) always divisible by the two numbers. Thus every multiple is divisible by the two numbers too.