Find the value of X

Number Theory Level 3

If 513X070 is a multiple of 7, then find the sum of the values of X which satisfy this condition.

(Do not cheat by using a Calculator)

Show your result algebraically.


The answer is 9.

Vijay Simha by Vijay Simha , 47, USA

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2 solutions

Vijay Simha
Aug 25, 2015

You can start by expressing:

513X070 = 4900000 + 230000 + 1000X + 70

= 7k1 + 210000 + 20000 + 1000X

= 7k2 + 21000 - 1000 + 1000X

= 7k3 + 1000X - 1000

= 7k3 + 1000(X - 1)

= 7k3 + 700(X - 1) + 300(X - 1)

= 7k4 + 300(X - 1)

= 7k4 + 280(X - 1) + 20(X - 1)

= 7k5 + 14(X - 1) + 6(X - 1)

= 7k6 + 6(X-1)

6(X-1) should be divisible by 7. which means X = 8 or X = 1.

Therefore the sum of the values of X which satisfy the condition = 8+ 1 = 9

2 Helpful 0 Interesting 0 Brilliant 0 Confused

So K(1,2,3,4,5,6) are just constants? It seems confusing to me.

Joeie Christian Santana - 5 years, 9 months ago

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Yes, As soon as you spot a multiple of 7, replace it by 7k1.

4900000 = 7x700000, So k1 = 700000

2100000 = 7x30000, so k2 = 700000 + 30000 and so on,

Vijay Simha - 5 years, 9 months ago

Why not? Kindly discuss that 1 also satisfying in the simple division method!

Palanisamy Jayavel - 5 years, 9 months ago 
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513X070 can be written as:
513(10**4) + X(10**3) + 70

We have to find X such that:
513(10**4) + X(10**3) + 70 = 0 ( mod 7 )
( 9 >= X >=0 )

70 = 0 ( mod 7 )

513 = 2 ( mod 7 )
10**4 = 4 ( mod 7 )
Then: 513(10**4) = 8 ( mod 7 ) = 1 ( mod 7 )

X(10**3) mod 7 must be 6 since 513(10**4) = 1 ( mod 7 )
10**3 = 6 ( mod 7 )
X = 1 ( mod 7 )

There are two possible values for x; which are 1 and 8

So the answer is 1+8 = 9

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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