Last visit was: 15 Sep 2024, 00:50 It is currently 15 Sep 2024, 00:50
Decision Tracker
My Rewards
Unanswered
Active Topics
New posts
New comers' posts
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Russ19
VP
VP
Joined: 29 Oct 2019
Posts: 1360
Own Kudos [?]: 1590 [2]
Given Kudos: 582
Send PM
If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi [#permalink] If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi   24 Sep 2021, 05:46
2
Bookmarks
00:00
A
B
C
D
E

Difficulty:

55% (hard)

Question Stats:

63% (02:10) correct 37% (02:18) wrong based on 76 sessions
Hide Show DIFFICULTY AND TIMER STATISTICS
If \(p\) is a positive integer less than\( 70\), and \(71^2 − 142p + p^2\) is divisible by \(49\), what is the remainder when \(p\) is divided by \(7\)?

A) 0

B) 1

C) 2

D) 4

E) 6
ShowHide Answer
Official Answer
B
User avatar
B
rishabh195
Intern
Intern
Joined: 06 Sep 2021
Posts: 16
Own Kudos [?]: 2 [0]
Given Kudos: 118
Send PM
Re: If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi [#permalink] If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi   24 Sep 2021, 06:14
sjuniv32
If \(p\) is a positive integer less than\( 70\), and \(71^2 − 142p + p^2\) is divisible by \(49\), what is the remainder when \(p\) is divided by \(7\)?

A) 0

B) 1

C) 2

D) 4

E) 6


I am not getting the answer as B. Please help

I am getting the value of p as 70, but because the value of p is less than 70..... I took it as 69 and when 69 divided by 7, it gives a remainder of 6.

But as per the OA, it should give the remainder of 1.

Where am I going wrong? Please help

Posted from my mobile device
User avatar
L
PyjamaScientist
Admitted - Which School Forum Moderator
Joined: 25 Oct 2020
Posts: 1124
Own Kudos [?]: 1125 [0]
Given Kudos: 629
Schools: Ross '25 (M$)
GMAT 1: 740 Q49 V42 (Online)
Send PM
Reviews Badge
Re: If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi [#permalink] If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi   24 Sep 2021, 08:06
rishabh195
sjuniv32
If \(p\) is a positive integer less than\( 70\), and \(71^2 − 142p + p^2\) is divisible by \(49\), what is the remainder when \(p\) is divided by \(7\)?

A) 0

B) 1

C) 2

D) 4

E) 6


I am not getting the answer as B. Please help

I am getting the value of p as 70, but because the value of p is less than 70..... I took it as 69 and when 69 divided by 7, it gives a remainder of 6.

But as per the OA, it should give the remainder of 1.

Where am I going wrong? Please help

Posted from my mobile device

How are you getting the value of P as 70, can you explain?


Here's how I went about it-

Given: 71^2 - 142p + p^2 is divisible by 49, which means, 71^2 - 142p + p^2 = 49k ---------(1)

If you rearrange 71^2 - 142p + p^2 , you can write it in the form of (a-b)^2.
That will be- (71 - p)^2

Using (1) above,
(71 - p)^2 = 49k
49k can be written as (7K)^2. Since the left side of the equation is a perfect square, the right side should be also a perfect square.

Equating both sides, you would get->

-> 71 - p = 7K
-> p = 71 - 7k

Now, the question is, what is the remainder when p is divided by 7.

-> 71/7 - 7K/7. Here, 7K/7 would yield no remainder. But 71/7 would yield a remainder of 1.

Hence, the option B is correct.
User avatar
G
SaquibHGMATWhiz
GMATWhiz Representative
Joined: 23 May 2022
Posts: 629
Own Kudos [?]: 514 [0]
Given Kudos: 6
Location: India
GMAT 1: 760 Q51 V40
Send PM
Re: If p is a positive integer less than 70, and 71^2 142p + p^2 is divi [#permalink] If p is a positive integer less than 70, and 71^2 − 142p + p^2 is divi   12 Jul 2023, 06:19
Expert Reply
Russ19
If \(p\) is a positive integer less than\( 70\), and \(71^2 − 142p + p^2\) is divisible by \(49\), what is the remainder when \(p\) is divided by \(7\)?

A) 0
B) 1
C) 2
D) 4
E) 6

Solution:
  • According to the question, \(p\) is a positive integer less than \(70\)
  • And \(71^2 − 142p + p^2=49k\) where k is an integer
    \(⇒(71-p)^2=49k\)
    \(⇒71-p=7\sqrt{k}\)
    \(⇒p=71-7\sqrt{k}\)
  • Remainder when \(p\) is divided by \(7=(\frac{71-7\sqrt{k}}{7})_r=(\frac{71}{7})_r-(\frac{7\sqrt{k}}{7})_r=1-0=1\)

Hence the right answer is Option B
GMAT Club Bot
gmatclubot
Re: If p is a positive integer less than 70, and 71^2 142p + p^2 is divi [#permalink]
12 Jul 2023, 06:19
Moderator:
Bunuel
Math Expert
95507 posts