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QUESTION: 1

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?

Solution:

QUESTION: 2

If *n *is a positive integer, what is the remainder when 3^{8n+3} + 2 is divided by 5?

Solution:

QUESTION: 3

What is the remainder when the positive integer *n* is divided by the positive integer *k*, where *k* > 1?

1) *n = (k+1) ^{3}*

2) *k* = 5

Solution:

QUESTION: 4

What is the remainder when the sum of the positive integers x and y is divided by 6?

1) When x is divided by 6, the remainder is 3. ?

2) When y is divided by 6, the remainder is 1.

Solution:

QUESTION: 5

When 20 is divided by the positive integer *k*, the remainder is *k *– 2. Which of the following is a possible value of *k*?

Solution:

QUESTION: 6

Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders?

1) The range of the seven remainders is 6. ?

2) The seven numbers selected are consecutive integers. ?

Solution:

**Correct Answer :- b**

**Explanation :** The trick here is to know that remainder is always non-negative integer less than divisor 0≤r<d0≤r<d, so in our case 0≤r<70≤r<7.

So the remainder upon division of any integer by 7 can be: 0, 1, 2, 3, 4, 5, or 6 (7 values).

(1) The range of the seven remainders is 6 --> if we pick 6 different multiples of 7 (all remainders 0) and the 7th number 6 (remainder 6) then the range would be 6 and the sum also 6. But if we pick 7 consecutive integers then we'll have all possible remainders: 0, 1, 2, 3, 4, 5, and 6 and their sum will be 21. Not sufficient.

(2) The seven numbers selected are consecutive integers --> ANY 7 consecutive integers will give us all remainders possible: 0, 1, 2, 3, 4, 5, and 6. It does not matter what the starting integer will be: if it's say 11 then the remainder of 7 consecutive integers from 11 divided by 7 will be: 4, 5, 6, 0, 1, 2, and 3 and if starting number is say 14 then the remainder of 7 consecutive integers from 14 divided by 7 will be: 0, 1, 2, 3, 4, 5 and 6. So in any case sum=0+1+2+3+4+5+6=21. Sufficient.

QUESTION: 7

When 15*n*, where *n* is a positive integer, is divided by 6, the remainder is *x*. What is the value of *x*?

1) When *n* is divided by 2, the remainder is 0. ?

2) When *n* is divided by 3, the remainder is 0. ?

Solution:

QUESTION: 8

If *n* is a positive integer, what is the remainder when (7^{(4n+3)})(6^{n}) is divided by 10?

Solution:

QUESTION: 9

For a nonnegative integer *n*, if the remainder is 1 when 2* ^{n} *is divided by 3, then which of the following must be true?

I. *n* is greater than zero. ?

II. 3* ^{n} *= (-3)

III. (√2)

Solution:

QUESTION: 10

If *n *is a positive integer and *r *is the remainder when (*n *– 1)(*n *+ 1) is divided by 24, what is the value of *r*?

1) 2 is not a factor of *n*. ?

2) 3 is not a factor of *n*.

Solution:

QUESTION: 11

What is the remainder when the two-digit, positive integer *x *is divided by 3 ?

1) The sum of the digits of *x *is 5.?

2) The remainder when *x *is divided by 9 is 5.

Solution:

QUESTION: 12

What is the remainder when the positive integer x is divided by 8?

1) When x is divided by 12, the remainder is 5. ?

2) When x is divided by 18, the remainder is 11. ?

Solution:

QUESTION: 13

When positive integer *x* is divided by positive integer *y*, the remainder is 9. If *x/y* = 96.12, what is the value of *y*?

Solution:

The remainder is 9 when x is divided by y, so x = yq + 9 for some positive integer q.

Dividing both side by y give x/y = q + 9/y

But x/y = 96.12 = 96 + 0.12

Equating the two expressions for x/y give q + 9/y = 96 + 0.12

Thus q = 96 and 9/y = 0.12

⇒ 9 = 0.12y

⇒ y = 9/0.12

⇒ y = 75

QUESTION: 14

For all positive integers *m *and *v*, the expression *m *Θ *v *represents the remainder when *m *is divided by *v*. What is the value of ( (98 Θ 33) Θ 17 )− ( 98 Θ (33 Θ 17) ) ?

Solution:

First for ((98 Θ 33) Θ 17), determine 98 Θ 33, which equalsd to 32, since 32 is remainder when 98 divided with 33 ( 98 = 2(33) + 32 ).

Then determine 32 Θ 17 which equals to 15, since 15 is the remainder when 32 is divided by 17( 33= 1(17) +15)

Thus ( (98 Θ 33) Θ 17 ) = 15

Next, for (98 Θ (33 Θ 17)) , determine 33 Θ 17, which equalsd to 16, since16is remainder when33 divided with17 ( 33 = 1(17) + 16 ).

Then determine 98 Θ 16 which equals to 2, since 2 is the remainder when 98 is divided by 16 ( 98 = 6(16) + 2)

Thus (98 Θ (33 Θ 17)) = 2

Finally ((98 Θ 33) Θ 17)− (98 Θ (33 Θ 17)) = 15 - 2 = 13

QUESTION: 15

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Solution:

1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49

A total of 17 numbers are there which gives a remainder 1 when divided by 3

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