How many 4-digit palindromic numbers are divisible by 7? Palindromic numbers are numbers that are symmetrical . Ex: 1001 , 2112 , 5005 and 3773 .

ant101
Apr 2, 2017

#3**+1 **

Lets see. So we have a 4-digit palindrome. that means the first digit and the last digit are the same. it also means the second digit and the third digit are the same. lets call the first one a, the second- b, thhrd- c, and fourth- d

a=d

b=c

we know that a*1000+b*100+c*10+d=a*1000+b*100+b*10+a=a*1001+b*110 is divisible by 7.

we know that 1001 is divisible by 7 (1001/7=143), and that 110 is NOT divisible by 7. that means we need b to be divisible by 7 (because a number divisible by 7+a number NOT divisible by7=a number NOT divisible by7

and

a number divisible by 7+a number divisible by 7=a number divisible by 7

the first digit of the number has to be between 1 and 9, and the other digits have to be betwee 0 and 9. meaning 1<=a<=9 and 0<=b<=9. if we need b to be divisible by 7 there are 2 options for b- 0 or 7. there are 9 options for a because a*1001 will always be divisible by 7 and 1<=a<=9 leaving 9 options for a.

2*9=18

so there are 18 4-digit palindromic numbers are divisible by 7

Ehrlich
Apr 3, 2017

#1**0 **

Well I am not pretty sure if it is correct but I think this:

1001 is divisible by 7, so we get 1001, 2002,..., til 9009 that is divisible by 7, so the only palindromes we can make from that is if we add 770, so we get a total of 9+9 that this is 18

So we have 18 numbers (1001, 1771,...)

I hope u understand it

LucyWhat
Apr 2, 2017

#2**0 **

There are 36 such integers:

-9779 | -9009 | -8778 | -8008 | -7777 | -7007 | -6776 | -6006 | -5775 | -5005 | -4774 | -4004 | -3773 | -3003 | -2772 | -2002 | -1771 | -1001 | 1001 | 1771 | 2002 | 2772 | 3003 | 3773 | 4004 | 4774 | 5005 | 5775 | 6006 | 6776 | 7007 | 7777 | 8008 | 8778 | 9009 | 9779 .

Guest Apr 2, 2017

#3**+1 **

Best Answer

Lets see. So we have a 4-digit palindrome. that means the first digit and the last digit are the same. it also means the second digit and the third digit are the same. lets call the first one a, the second- b, thhrd- c, and fourth- d

a=d

b=c

we know that a*1000+b*100+c*10+d=a*1000+b*100+b*10+a=a*1001+b*110 is divisible by 7.

we know that 1001 is divisible by 7 (1001/7=143), and that 110 is NOT divisible by 7. that means we need b to be divisible by 7 (because a number divisible by 7+a number NOT divisible by7=a number NOT divisible by7

and

a number divisible by 7+a number divisible by 7=a number divisible by 7

the first digit of the number has to be between 1 and 9, and the other digits have to be betwee 0 and 9. meaning 1<=a<=9 and 0<=b<=9. if we need b to be divisible by 7 there are 2 options for b- 0 or 7. there are 9 options for a because a*1001 will always be divisible by 7 and 1<=a<=9 leaving 9 options for a.

2*9=18

so there are 18 4-digit palindromic numbers are divisible by 7

Ehrlich
Apr 3, 2017