Çözüldü Divisibility and Remainder

The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

A) 45
B) 53
C) 72
D) 85
E) 100

Solution Without Needing To Check All Choices:
For 14 as "Divisor" and x as the "Quotient"; m = 14·x + q + 7....(I)
For 7 as "Divisor" and y as the "Quotient"; m = 7·y + q....(II)
Subtracting (I) and (II) side by side, x and y as natural numbers having the relation y = 2x + 1, for x ≥ 2 ⇒ y = {5, 7, 9, ...} and
1 ≤ q ≤ 6;
x-------y-------q ≤ 6-------m = 7·y + q
2-------5---------6---------------43
3-------7---------6---------------55
3-------7---------4---------------53.

Original Solution In The Book:

https://i.ibb.co/CPnYN5B/GRE.png
http://62.182.86.140/main/273000/2ababdb40db359595ee1949e765fb6d4/Jeff Kolby, Derrick Vaughn - GRE Math Bible-Nova Press (2008).pdf
Page 25, Question 5, Solution on page 27