Question

A number X is converted to base 7 and becomes a four-digit number. Its leftmost digit is removed and placed at the right of the remaining three digits, resulting in a new number in base 7 which is twice the value of X. Find the decimal representations of all such numbers X.

Answer 1

If the base-7 number is abcd, then
343b+ 49c + 7d + a = 2(343a + 49b + 7c + d)
So, you know that d = 137a-49b-7c
and 1 <= a,d <= 6
0 <= b,c <= 6
more later ...

Answer 2

please send me the rest thankyou

Answer 3

1<=a,b<=6
0<=c,d<=6.
as d=137a-49b-7c
if b=1, there is no chance for d to be one digit, same as b>2.
Thus b=2, then a=1, c=5, d=4. so X=(1254)7=480
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