Determine-the-number-of-five-digit-integers-37abc-in-base-10-such-that-each-of-the-numbers-37abc-37bca-and-37cab-is-divisible-by-37-

Question Number 19637 by Tinkutara last updated on 13/Aug/17

Determine the number of five - digit

integers (37 a b c) in base 10 such that

each of the numbers (37 a b c), (37 b c a)

and 37 c a b is divisible by 37 .

Answered by Rasheed.Sindhi last updated on 14/Aug/17

37 abc = 37000 + abc

37 ∣ 37000 + abc

37 ∣ 3700 \Rightarrow 37 ∣ abc

\Rightarrow abc is multiple of 37

As a may be 0, abc may be

2 - digit or 3 - digit multiple

of 37 :

37 \times 0, 37 \times 1, 37 \times 2, \dots, 37 \times n

000, 037, 074, \dots, 37 n ⩽ 999 : 37 n

is last term GP .

n = [\frac{999}{37}] = 27

\overset{1}{000}, (\overset{n = 27}{037, 074, \dots, 37 n} ⩽ 999)

1 + 27 = 28

Total multiples are 28 having

property given in the question .

For example

37 = 037 \Rightarrow abc = 037 = 37 \times 1

bca = 370 = 37 \times 10, cab = 703 = 37 \times 19

So number 37037

Its variations : 37370 & 37703

37 ∣ 37037, 3 ∣ 37370, 37 ∣ 37703

Commented by Tinkutara last updated on 14/Aug/17

Thank you very much Sir!