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## Threes and Sevens

Math brain teasers require computations to solve.

 Puzzle ID: #28470 Fun: (2.18) Difficulty: (2.83) Category: Math Submitted By: shenqiang Corrected By: kiss

Which is the smallest natural number that satisfies the following conditions:

(1) Its digits consist of, and only of 3's and 7's.
(2) Its digital sum is divisible by 3 and 7.
(3) The number itself is divisible by 3 and 7.

From conditions 1 and 2, we know that we need at least 3 7's and 7 3's.

Their sum is divisible by 3, therefore any number they make is divisible by 3.

By trial and error, we see that none of 3333333777, 3333337377, 3333337737, 3333337773, 3333373377, 3333373737, 3333373773, 3333377337 and 3333377373 is divisible by 7.

3333377733 is divisible by 7 (the quotient is 476196819.)

Therefore, the number we're looking for is 3333377733.

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 cms271828 Feb 06, 2006 Please stop sending these puzzles, especially when you don't make it clear if a 3 and a 7 are both mandatory. Theres not much fun in guessing numbers teen_wiz Feb 06, 2006 Wow. My brain's spinning. wolfmanh Feb 06, 2006 GREAT teaser, although I am not to good with these WJSMama Feb 06, 2006 I totally don't get these. paul726 Feb 08, 2006 I thought you did a great job! Jimbo Feb 10, 2006 cms, for someone who claims to be a Maths teacher, you don't seem to be very good at it. It says the digits consist of only 3's and 7's. How does that not tell you that there are madatory 3's and 7's. Like the number 33 consists of 3's and 7's ? Huh? I liked the puzzle. Well explained too! keveffect1 Feb 17, 2006 Now that I have done a couple these puzzles are very easy so don't worry about all of these negative comments they are people with limited mathematical skill stil May 10, 2006 It would be nice if hint pointed to easiest of these questions, so we could try it first. tonjawithaj Aug 07, 2006 I got stuck...I'm not sure what a digital sum is... my brain is off today... Johnny Nov 15, 2008 Nice! javaguru Jan 28, 2009 Silly me. Since it was not clear that the number had to have both 3s and 7s, I went with 777. Too bad since I would have enjoyed working out the intended solution. tonjawithaj Jan 28, 2009 Why isn't it clear that the number has to have 3s and 7s? #1 says "Its digits consist of, and only of 3's and 7's." The "and" would seem to me to be very clear. It doesn't say "or" or "and/or". jaysenf Oct 03, 2012 So it seems that 373737 also works and is less than the answer given. That's what we are looking for right?