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Just found out the mistake that I was making. t= 6, t = 13 satisfy both stmts but there are some additional values that satify stmt 2 (e.g. t= 1,t=8) and so we dont get the same remainder for all values of t so stmt 2 is out. And should be A

Just found out the mistake that I was making. t= 6, t = 13 satisfy both stmts but there are some additional values that satify stmt 2 (e.g. t= 1,t=8) and so we dont get the same remainder for all values of t so stmt 2 is out. And should be A

Is there a way to avoid listing all the numbers? because what if we miss out on a number that can be crucial to our decision like the mistake that you made? By the way, in statement 1, you will get 2 values if you try t= 6 and 20. As for statement 2, t=1 & 8 will still give you the remainder of 5. Plug them into (t+3)(t+2)/7

is there a generic equation that we can just use and avoid listing down all the numbers? because sometimes we might have to list A LOT of numbers until we can arrive to a confirmed answer. I'm starting to hate listing down numbers. Any other faster approach?

Just found out the mistake that I was making. t= 6, t = 13 satisfy both stmts but there are some additional values that satify stmt 2 (e.g. t= 1,t=8) and so we dont get the same remainder for all values of t so stmt 2 is out. And should be A

Is there a way to avoid listing all the numbers? because what if we miss out on a number that can be crucial to our decision like the mistake that you made? By the way, in statement 1, you will get 2 values if you try t= 6 and 20. As for statement 2, t=1 & 8 will still give you the remainder of 5. Plug them into (t+3)(t+2)/7

is there a generic equation that we can just use and avoid listing down all the numbers? because sometimes we might have to list A LOT of numbers until we can arrive to a confirmed answer. I'm starting to hate listing down numbers. Any other faster approach?

Yes.

(1) t = 7k + 6 (where k is integer) t^2 + 5t +6 = 49k^2 + 14k + 36 + 35k + 30 + 6 = 49k^2 + 14k + 72 (the first two terms are multiple of 7) so the reminder is equal to the remainder when 72 divided by 7, i.e. 2 (1) is sufficent

(2) t^2 = 7i +1 (where k is integer) t^2 + 5t +6 = 7i + 1 + 5t + 6 = 7i + 7 + 5t (the first two terms are multiple of 7) so the reminder is equal to the remainder when 5t divided by 7

Now we try to find out the remainder when 5t divided by 7: when i = 0, t^2 = 1, t = 1, remainder of t when divided by 7 = 1 when i = 5, t^2 = 36, t = 6, remainder of t when divided by 7 = 6 so, it give different results. (2) alone is not suff.

When dealing with remainders you only have to check the numbers that are in the set of possible remainders. In this problem that set is {0, 1, 2, 3, 4, 5, 6}.

Since 6^2/7 has a remainder of 1, we know all numbers that have a remainder of 6 will have a remainder of 1 when squared.

Since 1^2/7 and 6^2/7 both have remainders of 1, we know that merely knowing the remainder of t^2/7 is not sufficient to determine the remainder of t/7

Spend about an hour reading about modular arithmetic and you'll see why this is true

Just found out the mistake that I was making. t= 6, t = 13 satisfy both stmts but there are some additional values that satify stmt 2 (e.g. t= 1,t=8) and so we dont get the same remainder for all values of t so stmt 2 is out. And should be A

Is there a way to avoid listing all the numbers? because what if we miss out on a number that can be crucial to our decision like the mistake that you made? By the way, in statement 1, you will get 2 values if you try t= 6 and 20. As for statement 2, t=1 & 8 will still give you the remainder of 5. Plug them into (t+3)(t+2)/7

is there a generic equation that we can just use and avoid listing down all the numbers? because sometimes we might have to list A LOT of numbers until we can arrive to a confirmed answer. I'm starting to hate listing down numbers. Any other faster approach?

Yes.

(1) t = 7k + 6 (where k is integer) t^2 + 5t +6 = 49k^2 + 14k + 36 + 35k + 30 + 6 = 49k^2 + 14k + 72 (the first two terms are multiple of 7) so the reminder is equal to the remainder when 72 divided by 7, i.e. 2 (1) is sufficent

(2) t^2 = 7i +1 (where k is integer) t^2 + 5t +6 = 7i + 1 + 5t + 6 = 7i + 7 + 5t (the first two terms are multiple of 7) so the reminder is equal to the remainder when 5t divided by 7

Now we try to find out the remainder when 5t divided by 7: when i = 0, t^2 = 1, t = 1, remainder of t when divided by 7 = 1 when i = 5, t^2 = 36, t = 6, remainder of t when divided by 7 = 6 so, it give different results. (2) alone is not suff.

Ans is A

I have a problem with your statement 2. You provided an equation of \(t^2 = 7i +1\). So it means:

\(7i+1 + 5sqrt(7i+1) + 6\) so we don't have 2 variables anymore, but just one because i plugged in your \(t^2 = 7i +1\) into \(t^2 + 5t + 6\), which finally means \(7i + 5sqrt(7i+1) + 7\), so our remainder for statement 2 should be zero because our constant 7 is divisible by 7. what's wrong with that?

Last edited by tarek99 on 07 Sep 2008, 06:34, edited 1 time in total.

our constant here is 12, which means that when you divide it by 7, the remainder should be 5. So what's wrong with this approach? I would appreciate it if someone can help me with this problem. regards,

is there a generic equation that we can just use and avoid listing down all the numbers? because sometimes we might have to list A LOT of numbers until we can arrive to a confirmed answer. I'm starting to hate listing down numbers. Any other faster approach?[/quote]

Yes.

(1) t = 7k + 6 (where k is integer) t^2 + 5t +6 = 49k^2 + 14k + 36 + 35k + 30 + 6 = 49k^2 + 14k + 72 (the first two terms are multiple of 7) so the reminder is equal to the remainder when 72 divided by 7, i.e. 2 (1) is sufficent

(2) t^2 = 7i +1 (where k is integer) t^2 + 5t +6 = 7i + 1 + 5t + 6 = 7i + 7 + 5t (the first two terms are multiple of 7) so the reminder is equal to the remainder when 5t divided by 7

Now we try to find out the remainder when 5t divided by 7: when i = 0, t^2 = 1, t = 1, remainder of t when divided by 7 = 1 when i = 5, t^2 = 36, t = 6, remainder of t when divided by 7 = 6 so, it give different results. (2) alone is not suff.

Ans is A[/quote]

I have a problem with your statement 2. You provided an equation of \(t^2 = 7i +1\). So it means:

\(7i+1 + 5sqrt(7i+1) + 6\) so we don't have 2 variables anymore, but just one because i plugged in your \(t^2 = 7i +1\) into \(t^2 + 5t + 6\), which finally means \(7i + 5sqrt(7i+1) + 7\), so our remainder for statement 2 should be zero because our constant 7 is visible by 7. what's wrong with that?[/quote]

Hi, the reason we have to to find out the possible number of t is that we do not know the remainder when t is divided by 7.

Can you tell the remainder when (7i+1)^(0.5) divided by 7?

On the other hand, why t^2 + 6 = 7i + 7, which is always divisible by 7, therefore remainder is 0.

So we need to find out the remainder when (7i+1)^(0.5) divided by 7 in order to find out the remainder when t^2 + 5t +6 is divided by 7.

So the next step is to insert diffenert possible value of i, to find t, to see whether it gives consistent answer.

Since t is a positive integer. when i = 0, t^2 = 1, t = 1, remainder of t when divided by 7 = 1 when i = 5, t^2 = 36, t = 6, remainder of t when divided by 7 = 6

So we only have variable answer, so not suff to solve the problem

our constant here is 12, which means that when you divide it by 7, the remainder should be 5. So what's wrong with this approach? I would appreciate it if someone can help me with this problem. regards,

The mistake is \(t^2 = 7k + 1\) is NOT the same as \(t = 7k + 1\)

Think about: if \(t^2 = 7*5 + 1\) \(t = 6\) when divided by 7, the remainder is 6, or \(t = 7i + 6\)

if \(t^2 = 7*0 + 1\) \(t = 1\) when divided by 7, the remainder is 1, or \(t = 7i + 1\)

our constant here is 12, which means that when you divide it by 7, the remainder should be 5. So what's wrong with this approach? I would appreciate it if someone can help me with this problem. regards,

The mistake is \(t^2 = 7k + 1\) is NOT the same as \(t = 7k + 1\)

Think about: if \(t^2 = 7*5 + 1\) \(t = 6\) when divided by 7, the remainder is 6, or \(t = 7i + 6\)

if \(t^2 = 7*0 + 1\) \(t = 1\) when divided by 7, the remainder is 1, or \(t = 7i + 1\)

cool, so even if i was wrong about \(t^2 = 7k + 1\) being the same as \(t = 7k + 1\). I still can create an equation, no? Here's statement 2 again:

\(t^2 = 7k + 1\), so if i plug it into the given equation \(t^2 + 5t + 6\), i would get:

\((7k + 1) + 5 sqrt(7k+1) + 6\), which is equal to \(7k + 5sqrt(7k+1) + 7\), so what's wrong with this? why aren't we considering the 7 in the last part of the addition to conclude that the remainder will be zero? 7 is the constant in this equation after all. Haven't we been picking the constant of every such formula all the time? why aren't we doing so here this time? That's my main question. regards,

cool, so even if i was wrong about \(t^2 = 7k + 1\) being the same as \(t = 7k + 1\). I still can create an equation, no? Here's statement 2 again:

\(t^2 = 7k + 1\), so if i plug it into the given equation \(t^2 + 5t + 6\), i would get:

\((7k + 1) + 5 sqrt(7k+1) + 6\), which is equal to \(7k + 5sqrt(7k+1) + 7\), so what's wrong with this? why aren't we considering the 7 in the last part of the addition to conclude that the remainder will be zero? 7 is the constant in this equation after all. Haven't we been picking the constant of every such formula all the time? why aren't we doing so here this time? That's my main question. regards,

: )

maybe you can answer your own questions by answering my questions below: is 14 + 7 divisible by 7? is 16 + 7 divisible by 7? If we can conclude the whole number is divisible by 7 by pulling out a contant 7, then all number can be divisble by 7. 99 = 92 + 7, then is 99 divisible by 7? the remainder is 0? sure not.

if x = A + B + C, if A and B are both divisible by 7 (i.e. remainder is 0), then we know that the remainder when C is divided by 7, is equal to the remainder when x divided by 7.

If A, B are not divisible by 7, we need to find out the remainder when they are divided by 7, in order to find the remainder of x when divided by 7.

Go back to your equation, very nice, 100% correct.

\(7k + 5sqrt(7k+1) + 7\)

A = 7K B = 7 A & B are divisible by 7, so remainder is 0

Then we can conclude that the remainder in question is equal to the remainder when \(5sqrt(7k+1)\) divided by 7.

Is it divisible by 7? yes or no. Clearly, we cannot arrive a certain yes, because we cannot pull out a factor of 7 from it, like this form \(7*5sqrt(7k+1)\)

In order to prove that no certain answer can be reached, we try to fill in some numbers for k.

After filling in the numbers, we can 100% conclude that the term \(5sqrt(7k+1)\) and therefore \(t^2 + 5t + 6\) may or maynot be divisible by 7 (varies remainder), depending the actual value of k (i.e. t)

cool, so even if i was wrong about \(t^2 = 7k + 1\) being the same as \(t = 7k + 1\). I still can create an equation, no? Here's statement 2 again:

\(t^2 = 7k + 1\), so if i plug it into the given equation \(t^2 + 5t + 6\), i would get:

\((7k + 1) + 5 sqrt(7k+1) + 6\), which is equal to \(7k + 5sqrt(7k+1) + 7\), so what's wrong with this? why aren't we considering the 7 in the last part of the addition to conclude that the remainder will be zero? 7 is the constant in this equation after all. Haven't we been picking the constant of every such formula all the time? why aren't we doing so here this time? That's my main question. regards,

: )

maybe you can answer your own questions by answering my questions below: is 14 + 7 divisible by 7? is 16 + 7 divisible by 7? If we can conclude the whole number is divisible by 7 by pulling out a contant 7, then all number can be divisble by 7. 99 = 92 + 7, then is 99 divisible by 7? the remainder is 0? sure not.

if x = A + B + C, if A and B are both divisible by 7 (i.e. remainder is 0), then we know that the remainder when C is divided by 7, is equal to the remainder when x divided by 7.

If A, B are not divisible by 7, we need to find out the remainder when they are divided by 7, in order to find the remainder of x when divided by 7.

Go back to your equation, very nice, 100% correct.

\(7k + 5sqrt(7k+1) + 7\)

A = 7K B = 7 A & B are divisible by 7, so remainder is 0

Then we can conclude that the remainder in question is equal to the remainder when \(5sqrt(7k+1)\) divided by 7.

Is it divisible by 7? yes or no. Clearly, we cannot arrive a certain yes, because we cannot pull out a factor of 7 from it, like this form \(7*5sqrt(7k+1)\)

In order to prove that no certain answer can be reached, we try to fill in some numbers for k.

After filling in the numbers, we can 100% conclude that the term \(5sqrt(7k+1)\) and therefore \(t^2 + 5t + 6\) may or maynot be divisible by 7 (varies remainder), depending the actual value of k (i.e. t)

oohhhh.....i see, i see...now i understand what's going on hear. Thanks a lot for the great explanation. I understand now. Thanks again

Another question I have for you. Let's say that x = a + b + c and we're trying to divide x by 7. Suppose that when a is divided by 7, the remainder is 2; when b is divided by 7, the remainder is 1; and when c is divided by 7, the remainder is 3. does that mean that the overall remainder when x is divided by 7 will be the addition of each of these remainders? so 2+1+3 = 6?

Another question I have for you. Let's say that x = a + b + c and we're trying to divide x by 7. Suppose that when a is divided by 7, the remainder is 2; when b is divided by 7, the remainder is 1; and when c is divided by 7, the remainder is 3. does that mean that the overall remainder when x is divided by 7 will be the addition of each of these remainders? so 2+1+3 = 6?

Hi tarek, seems you totally understand now.

Yes, the remainder is 6.

To elaborate more, (actually, the method is very similar to what we used above)

Suppose that when a is divided by 7, the remainder is 2; when b is divided by 7, the remainder is 1; and when c is divided by 7, the remainder is 3

then we can let: a = 7q + 2 b = 7z + 1 c = 7i + 3

then x = 7*(q+z+i) + 6 so the remainder is 6

let's look at one more example: A = 7Q + 6 B = 7Z + 6 C = 7I + 5 then X = 7*(Q+Z+I) + 17 So remainder is equal to the remainder when 17 divided by 7, i.e. 3

if you still confused, think about then X = 7*(Q+Z+I) + 7 + 10 then only the term 10 is not divisble by 7 so the remainder is equal to 10 divided by 7, i.e. 3

Another question I have for you. Let's say that x = a + b + c and we're trying to divide x by 7. Suppose that when a is divided by 7, the remainder is 2; when b is divided by 7, the remainder is 1; and when c is divided by 7, the remainder is 3. does that mean that the overall remainder when x is divided by 7 will be the addition of each of these remainders? so 2+1+3 = 6?

Hi tarek, seems you totally understand now.

Yes, the remainder is 6.

To elaborate more, (actually, the method is very similar to what we used above)

Suppose that when a is divided by 7, the remainder is 2; when b is divided by 7, the remainder is 1; and when c is divided by 7, the remainder is 3

then we can let: a = 7q + 2 b = 7z + 1 c = 7i + 3

then x = 7*(q+z+i) + 6 so the remainder is 6

let's look at one more example: A = 7Q + 6 B = 7Z + 6 C = 7I + 5 then X = 7*(Q+Z+I) + 17 So remainder is equal to the remainder when 17 divided by 7, i.e. 3

if you still confused, think about then X = 7*(Q+Z+I) + 7 + 10 then only the term 10 is not divisble by 7 so the remainder is equal to 10 divided by 7, i.e. 3

perfect. well understood. Thanks a lot for your explanation. I totally understand now.

i've also noticed something interesting. For example:

14 = 5 + 9

suppose we're trying to divide 14 by 7. 14 is divisible by 7. However, when you separately divide 5 by 7 and 9 by 7, non of them are divisible by 7. But when the sum of their remainders are a multiple of 7, then the whole number is divisible by 7. For example: 5 divide by 7, the remainder is 5. When you divide 9 by 7, the remainder is 2. Therefore, 5+2 = 7, which is a multiple of 7, so 14 is divisible by 7. Is that a correct way to approach such a situation?

i've also noticed something interesting. For example:

14 = 5 + 9

suppose we're trying to divide 14 by 7. 14 is divisible by 7. However, when you separately divide 5 by 7 and 9 by 7, non of them are divisible by 7. But when the sum of their remainders are a multiple of 7, then the whole number is divisible by 7. For example: 5 divide by 7, the remainder is 5. When you divide 9 by 7, the remainder is 2. Therefore, 5+2 = 7, which is a multiple of 7, so 14 is divisible by 7. Is that a correct way to approach such a situation?

Yes, this is correct. Good suggestion.

You may also want to think about the following question.

What is the remainder when n is divided by 9? n is a positive integer.

(1) the sum of all digits of n is divisible by 11.